Copyright: Copyright ® 1999, Physicon Ltd


Contents

1. Introduction

What is OPEN PHYSICS

Brief Simulations Description

2. Working with OPEN PHYSICS

Program Operation

Choosing the Simulation

3. Physics Simulations

List of Physics Simulations

Choosing physical parameters

Video with Physical Experiment

Simulations Operation

Physics Description

Voice Description

Problems and Solutions

4. Tools

Calculator

Notepad


What is OPEN PHYSICS

OPEN PHYSICS includes comprehensive Multimedia courses in Physical Sciences based upon modern innovative software architectures and technologies (OLE/OCX, Netscape plug-ins). It allows to use the course in open educational systems and Internet distance learning systems.

OPEN PHYSICS consists of more than hundred simulations of physical phenomena, video laboratory examples. There are a lot of questions and problems in the course. Interactive dialogue with user and data visualisation of physical experiments allow students to explore the physical sciences in depth.


Brief Simulations Description

1. Electricity and Magnetism

2. Optics

3. Modern Physics


Electricity and Magnetizm. Brief Simulations Description.
Electricity and Magnetism. Brief Description.

1. Electric Field of Point Charges. The computer simulation demonstrates lines of force and equipotential surfaces of the electric fields of one or two point charges. It allows to change the magnitudes and signs of the charges and the distance between them and to measure the electric field intensity and the potential at every point on the screen.

2. Interaction of Electric Charges. The computer program allows to create on the screen different configurations of three point charges and to measure the force if interaction between them. The simulation illustrates the principle of superposition of electric fields.

3. Parallel - Plate Capacitor. This program allows to show on the screen the lines of force and equipotential surfaces of the electric field of a parallel - plate capacitor carried different charges on the plates. The simulation demonstrates the uniform electric field inside the capacitor and effect of fringing of the field at the edges of the plates. If is possible to measure the force acting on the test point charge inside and outside the capacitor.

4. Direct Current Circuits.  The simulation allows to create on the screen different direct current circuits consisting of batteries and resistors. It is possible to change the magnitudes of EMF of the butteries and resistance of the resistors and to put ammeters and voltmeters in the circuit. The program illustrates Ohms law and Kirchhoffs rules for complex branching circuits.

5. DC Circuits with Capacitors. The program permits to build on the screen different circuits consisting of butteries, capacitors and bulbs instead of resistors. It is possible to observe the relaxation processes and to measure the changing of the currents and voltages with time. Besides, the simulation illustrates the rules of determining of total capacitance of parallel or series junction of capacitors.

6. Magnetic Field of a Straight Conductor. The computer model allows to demonstrate lines of magnetic field of a straight conductor with different magnitudes of the current and to measure the magnetic field at every point on the screen. There is the mode of the simulation in which the structure of the magnetic field is qualitatively illustrated as in real experiment with the aid of iron filings.

7. Magnetic Field of a Circular Loop. This simulation enables to observe lines of force of magnetic field of circular loop with different values of the current in it and to measure the magnetic field along the axis of the loop. It is possible to carry out the qualitative experiment to observe the structure of the magnetic field with the aid of iron filings.

8. Magnetic Field of a Solenoid. The program enables to observe lines of force of magnetic field of a solenoid and to measure the magnetic field along its axis with different values of the current. It is possible to observe qualitatively the structure of the magnetic field of the solenoid with the aim of iron filings.

9. Interaction of Two Parallel Currents The simulation allows to study the magnetic interaction of two parallel currents. It enables to change the magnitudes and directions of the currents and the distance between conductors. In particular, one can carry out the computer experiment on interaction of two currents, each equal to one ampere, placed at the distance of one meter. This experiment is the basis of the definition of the fourth basic unit in SI - the ampere.

10. Torque on a Current - Carrying Loop.This program demonstrates the arising of the torque on a current - carrying loop under the action of magnetic field and allows to measure the torque at various conditions.

11. Motion of a Charged Particle in Electric Field. One can observe the motion of an electron in uniform electric field inside the parallel - plate capacitor It is possible to vary the intensity of the electric field, the magnitude and the direction of initial velocity of the electron. The trajectory of the particle, its coordinates and components of the velocity with time are represented on the screen.

12. Motion of a Charged Particle in Magnetic Field. The program allows to study the motion of a charged particle in the uniform magnetic field. It shows that magnetic force is always perpendicular to the velocity of the particle and hence the magnetic force cant do work on the particle. One can calculate in this computer experiment the radius of the circular trajectory of the particle in the uniform magnetic field and the period of one cycle.

13. Velocity Selector. The program illustrates the performance of the simplest velocity selector in which charged particles travel along straight line in crossed electric and magnetic fields. It is possible to choose different particles for the experiment (for example, electron, proton or [image] - particle) and to change components of theirs speed as well as the magnitudes of electric and magnetic fields.

14. Mass Spectrometer. The computer simulation enables to study the action of the device, which is used to separate isotopes and for measurements of relative charges of the particles. It is possible to choose two isotopes of some chemical element for the computer experiment and to vary the magnitudes of speed and magnetic field. The difference between radii of circular trajectories of these isotopes is displayed.

15. Faraday Experiment I.The program illustrates a number of qualitative Faradays experiments on the electromagnetic induction in moving and still conductors.

16. Faraday Experiment II. The program illustrates a number of qualitative Faradays experiments on the electromagnetic induction in moving and still conductors.

17. Motion Electromotive Force  The simulation represents the computer experiment on electromagnetic induction in moving conductor. One can vary the speed of the conductor, its size, electrical resistance and the magnitude of the magnetic field. The magnitude of the EMF and the current in the circuit are displayed on the screen.

18. The Alternating Current Generator.The program illustrates the work of the altering current generator, invented by Faraday. It is possible to change the speed of rotation of the rectangular frame and the magnitude of the magnetic field. The graphs of the magnetic flux and EMF are displayed.

19. The R - C Circuits.The program allows to study the process of discharging of the capacitor through the resistor. One can vary the initial charge on the capacitor, the values of the capacitance and the resistance. The changing of the charge with time is displayed graphically on the screen.

20. The R - L Circuits.The simulation allows to study the process of increasing of the current in the circuit consisting of a coil with inductance, a resistor and a buttery, connected in series. One can change the values of the inductance, the resistance and the ENF of the battery. The dependence of the current on time is shown graphically.

21. Free Oscillations in L-R-C Circuits. The program demonstrates free oscillations in RLC circuit. It is possible to vary the circuit parameters. The graphs of the capacitor voltage and the current in the circuit versus time are shown. One can observe the process of transition of the electrical energy of the capacitor into magnetic energy of the coil and back.

22. Forced Oscillations in L-R-C Circuits.  The simulation allows to study the processes in RLC circuit under the action of the external harmonic voltage from generator. It is possible to vary circuit parameters and the frequency of the generator. The graph of the resonance curve is displayed.


Optics. Brief Simulations Description.

Optics. Brief Description.

1. Reflection and Refraction. The simulation allows to study the laws of the reflection and refraction. One can vary the angle of incidence and the index of refraction of the media. The reflected and refracted rays are shown on the screen as well as values of corresponding angles. It is possible to observe the phenomenon of total internal reflection.

2. The Plane Mirror. The program illustrates the appearance of the virtual image of an object made by plane mirror.

3. The Spherical Mirror.  The program shows the rays path, reflected by spherical mirror and the appearance of a real or virtual images. One can vary the position of an object and the radius of curvature of convex or concave mirror.

4. The Lens  The program shows the rays path in converging and diverging lenses with different optical power and the appearance of a real or virtual images. One can vary the position of the object and the optical power of the lens. The concept of focal point, focal length and optical power of thin lens are illustrated.

5. The System of Two Lenses. The program allows to construct the image of the object in optic system consisting of two lenses. The position of an object, optical powers of both lenses and the distance between them cam be changed. In particular one can build the models of telescope and microscope.

6. The Eye. The simulation illustrates the path of optical rays in the eye and the position of the image relative to retina depending on the position of the object. It is possible to choose type of the eye - normal, nearsighted or farsighted and to look through the phenomenon of eye accommodation. It is also possible to study glasses operation.

7. The Telescope. The program allows to study the operation of the astronomical telescope consisting of two converging lenses. One can choose the focal lengths of the objective and the eyepiece and get acquainted with the notion of angular magnification.

8. The Microscope. The simulation allows to get acquainted with the microscope operation. One can change optical powers of the objective and eyepiece and calculate the magnification of the microscope.

9. Newton's Rings. The simulation allows to observe circular fringes formed by interference of monochromatic light reflected from two surfaces of thin film between converging lens and plane glass plate. It is possible to vary wavelength of light and radius of curvature of lens surface and to measure radii of dark rings.

10. Interference Experiment of Young. The program allows to study the phenomenon of interference on the example of Youngs experiment. It is possible to observe interference fringes on the screen, plot of light intensity distribution and determine the width of interference fringes. The simulation allows to vary light wavelength, the distance between two splits in Youngs scheme and to observe the influence of these parameters on interference pattern.

11. Diffraction. The computer program allows to study the phenomenon of diffraction of light on round obstacles (a disk or a aperture) and one one-dimensional obstacles (a slit or a needle). One can get acquainted with the notion of Frenels zones. It is possible to vary the sizes of the obstacles and wavelength of light.

12. Fresnel's Zones. The simulation enables to observe diffraction patterns formed by diffraction of light on round apertures and round disks. One can get acquainted with the notion of Frenels zones and the operation of so called Frenels zone plates. It is possible to change the wavelength of light.

13. The Diffraction Grating.The program enables to study the operation of the diffraction grating as spectral tool. It is possible to vary the period of grating and wavelength of incident light.

14. Circular Apertures and Resolving Power.The program allows to observe the diffraction patterns in the focal plane of the objective of telescope due to the diffraction of light on the lens circular aperture and to study the phenomenon of the diffraction limit of resolving power of the optical instrument. It is possible to vary the wavelength of light, the diameter of the aperture and the angle between two point sources.

15. Polarizers. The simulation allows to get acquainted with the Malus law when unpolarized light passes trough two consequent polaroids.

16. Polarization of Light. The simulation represents the kinematic model of elliptically polarized light. It is shown that in general case polarized light can be considered as a superposition of two waves linearly polarized in mutually perpendicular planes. The cases of linear, circular and elliptical polarization of light are considered.


Modern Physics. Brief Simulations Description.

Modern Physics. Brief Description.

1. The Photoelectric Effect.  The program allows to study the photoelectric effect. It is possible to vary a lot of parameters - the wavelength and intensity of the incident light, magnitude and sign of the voltage between the anode and photocathode. The program allows to get through the concepts of stopping potential and threshold frequency.

2. The Compton Scattering. The simulation allows to study the phenomenon of x - rays scattering on free electrons. One can change the parameters of the computer experiment, namely the wavelength of incident radiation and the angle of scattering. The wavelength of the scattered and the plot of intensity distribution versus wavelength are displayed on the screen.

3. The Bohr's Postulates.  The simulation is intended for studying the quantum properties of the atomic system. One can get through the notions of energy levels and stationary Bohrs orbits of hydrogen atom and the transition between energy levels. Bohr frequency rule is studied.

4. Standing Electron Waves on a Circular Orbit.  The program allows to get acquainted with Bohr quantum postulates and de Brogles theory of existing standing electron waves on stable circular orbits of hydrogen atom.

5. The Wave Properties of Particles.  The program represents the computer experiment of the electron beam passing through one or two slits. It allows to get acquainted with duality of the microobjects and the existing of wave and corpuscular properties simultaneously. The Heisenberg uncertainty principle is illustrated.

6. The Electron Diffraction. This simulation concerns with the basic notion of the quantum physics - the notion of duality of all material objects. It illustrates the phenomenon of scattering of electrons in crystals by means of the mental experiment of electron diffraction on grating. It shows that the diffraction pattern appear as the result of the probability process. One can change the period of the diffraction grating, the speed of electrons and observe the appearance of the diffraction maxima. The wavelength of electrons and the plot of intensity distribution are displayed on the screen.

7. The Laser. Two Levels Model.  The simulation represents the model of different processes taken place in two - level quantum system when light beam is propagating: absorption of photons, spontaneous and stimulated emissions. One can get through the concept of pumping, inversion population and amplifier of light.

8. The Binding Energy of Nucleus.  The simulation allows to study the notions of binding energy of nucleus and defect of mass. The plots of dependence of number of neutrons versus number of protons in stable nucleus and the dependence of binding energy per nucleon versus mass number are represented on the screen. One can pick out the number of neutrons and protons to build stable nuclei and determine the binding energy per nucleon.

9. Relativity of Time.  The program allows to get acquainted with one of important consequences of Einsteins special theory of relativity - the relativity of time. One can observe the experiment on measuring the time interval between two events from the point of view of two observes in different frames of reference.

10. Relativity of Length. The simulation allows to study two postulates of Einsteins special theory of relativity - the principle of relativity and the principle of constancy of the speed of light. One can carry out the experiment on measuring the length of the ruler from the point of view of two observes in different frames of reference. It is possible to change relative speed of frames. Result of measuring the length of the ruler in moving frame of reference is displayed.


Program Operation

The program provide user to operate simulations and customise simulations, their properties and windows

The program contains following toolbar

Brief description of buttons

 - Exit

 - Make Simulations Navigator

 - Pause Simulation

 - Execute Simulation in step-by-step mode

 - Continue Simulation

 - Run All Executed Simulation

 - Run Simulation in Active Window only

 - Stop currently playing sounds

 - Show Information About Product


Choosing the Simulation

You can choose simulation in Simulations Navigator.

To create Simulations Navigator you should choose item "Make Simulations Navigator" in menu "View" or click button

on toolbar. After this you will obtain nonstructured list.

To run simulation from Simulations Navigator you should double click simulation icon.


Physics Description of Simulations
List of Physics Simulations

1. Electricity and Magnetizm

1. Electric Field of Point Charges.

2. Interaction of Electric Charges.

3. Parallel - Plate Capacitor.

4. Direct Current Circuits.

5. DC Circuits with Capacitors.

6. Magnetic Field of a Stright Conductor.

7. Magnetic Field of a Circular Loop.

8. Magnetic Field of Solenoid.

9. Interaction of Two Parallel Currents

10. Torque on a Current - Carrying Loop.

11. Motion of a Charged Partcle in Electric Field.

12. Motion of a Charged Partcle in Magnetic Field.

13. Velocity Selector.

14. Mass Spectrometer.

15. Faraday Experiments I.

16. Faraday Experiments II.

17. Motion Electromotive Force

18. The Alternating Current Generator.

19. The R - C Circuits.

20. The R - L Circuits.

21. Free Oscillations in L-R-C Circuits.

22. Forced Oscillations in L-R-C Circuits.

2. Optics

1. Reflection and Refraction.

2. The Plane Mirror.

3. The Spherical Mirror.

4. The Lens

5. The System of Two Lenses.

6. The Eye.

7. The Telescope.

8. The Microscope.

9. Newton's Rings.

10. Interference Experiment of Young.

11. Diffraction.

12. Fresnel's Zones.

13. The Diffraction Grating.

14. Circular Apertures and Resolving Power.

15. Polarizers.

16. Polarization of Light.

3. Modern Physics

1. The Photoelectric Effect.

2. The Compton Scattering.

3. The Bohr's Postulates.

4. Standing Electron Waves on a Circular Orbit.

5. The Wave Properties of Particles.

6. The Electron Diffraction.

7. The Laser. Two Levels Model.

8. The Binding Energy of Nucleus.

9. Relativity of Time.

10. Relativity of Length.


Choosing Physical Parameters

The simulations contain following interface elements to choose physical parameters, run processes or change their input parameters. You can use both mouse and keyboard to operate simulation controls.

Command Buttons

This control is used to perform some actions like run or stop simulation, reset parameters and so on (depending on the text on the button). It works like common Windows button. You can operate it using mouse or keyboard.

Floating bands

This control is used to choose various continuous physical parameters in simulations. You can use the buttons with arrows or drag the ruler and thumb. The value in the window displays the parameter's current value. You can operate it using mouse or keyboard (arrow keys, PgUp, PgDn, Home, End).

Thermometers

This control is used to choose a temperature. You can use the buttons with arrows or drag the thumb. The value in the window displays the chosen temperature. You can operate it using mouse or keyboard (arrow keys, PgUp, PgDn, Home, End).

Radio buttons

This control is used to choose the case from the given ones. It works like common Windows radiobutton group. You can operate it using mouse or keyboard.

Check Boxes

This control is used to set/reset switches (options). It works like common Windows check box button. You can operate it using mouse or keyboard.

Wave band

This control is used to choose light wavelength (and, so, the colour). You can use the buttons with arrows or click at colour area or thumb position. The value in the window displays the chosen wavelength, the thumb points chosen colour. You can operate it using mouse or keyboard (arrow keys, PgUp, PgDn, Home, End).

Drag-and-drop

To change some object's position, drag the object to some location by drag-and-drop technique. When the cursor changes on

you can drag the object with pressed left mouse button.

Point-and-click

To point object's position, the point-and-click technique is used. When the cursor changes on

you can point the new objects location by left-click.


Simulation Operation

You can stop all running simulations choosing "Pause all" in "Simulation" or using button

If simulation are stopped you can observe simulations step by step using "Do step" in "Simulation" or button

To continue stopped simulations you should choose "Continue all" in "Simulation" or using button

You can Run all simulations simultaneously or only active one.

To run all simulation choose "Run All" in "Simulation" or use button

To stop all running simulation except active one choose "Run Active Only" in "Simulation" or use button

This is useful for not very powerful computers.


Physics Description

You can read physics description of simulations and videos of physical experiments. To call it choose use button

Then Physics Description will be presented as Windows Help.


Voice Description

(is not available in Light Edition version)

If you have sound card you can listen physical description of simulation and video with physical experiment. To play sound you should use button

If you want to stop playing sound, click the following button at program toolbar


Video Description

(is not available in Light Edition version)

The program contains a set of movie with physical experiment. To play video from Simulations Navigator you should double click video icon


Problems and Solutions

The program contains a set of problems for simulations. To run problem you should use button

Then the window with problem conditions, input window, buttons Test, Close and Answer will appear.

User should enter the answer as numerical value in input window. To test answer with correct one user should click button Test.

To show solution user should click Answer button.

To exit problem user should click Close button.

How to input a problem solution

The problem enables you to enter your answer of the problem in the form of numerical value.

The physical units suggested for the answer are indicated.

The examples of numbers representation:

123,          0.123,          8.31


Calculator

To run Windows Calculator you should choose Calculator in Tools.


Notepad

To run Windows Notepad you should choose "Notepad" in "Tools"


Electric Field of Point Charges

According to modern physical conceptions, in the space around electric charges, appears an electric field. The field of charges at rest is called electrostatic.

Charges in the electric field are acted upon by a force.

The ratio of the force acting on the charge in the given field point to the value of the charge is called the electric field intensity or simply electric field

The electric field intensity is a vector quantity. The direction of the vector [image] coincides with the direction of the force acting on a positive test charge.

The magnitude of the electric field caused by a point charge q at a distance r from the charge is

,

where [image] [image] is electric constant. Such a field is called Coulomb's field.

The electric field caused by a set of charges satisfies the superposition principle:

,

where [image]...are the fields caused by the individual point charges.

For the graphic representation of the electric field, electric field lines are used. They are drawn so that at every point the direction of the electric field [image] is tangent to the electric field line at that point. Electric field lines are drawn so that their spacing is inversely proportional to the magnitude of the electric field.

Potential is an energetic characteristic of a field. Potential at a given point equals the work of electric forces to move a unit positive charge from this point to infinity. Potential is a scalar quantity.

The potential of a Coulomb's field caused by a point charge q is

An equipotential surface is defined as a surface on which the potential is the same at every point. Field lines are always perpendicular to equipotential surfaces.


Interaction of Electric Charges

The law of interaction of point charges was experimentally found by Charles de Coulomb in 1784. In SI Coulomb's law can be expressed as

,

where [image] [image] is electric constant, [image] and [image] are the magnitudes of point charges, and r is the distance between them. The direction of the force [image] is along the line connecting the charges. It is a repulsive force if the signs of [image] and [image] are the same [image], and attractive if they are opposite [image].

An interaction of electric charges that are at rest is called electrostatic.

The electrostatic interaction satisfies the principle of superposition. If a charge [image] interacts simultaneously with the charges [image] and [image], the total force [image] acting on that charge is the vector sum of the forces [image] and [image] that the two charges [image] and [image] would exert individually on [image]:

.


Parallel - Plate Capacitor

A parallel-plate capacitor consists of two parallel conducting plates, separated by a distance d.

This distance is usually small in comparison with their dimension. Nearly all the field of such a capacitor is localized in the region between the plates. In this region the field is approximately uniform and equals

.

Here [image] is the charge of each plate, [image] is the area of the plate. The ratio [image] is the magnitude of surface charge density of the plates.

If one puts a point charge [image] in the region between the plates, it will be acted upon by a force

.

If [image], the direction of the force coincides with the direction of the field lines; and if [image], the direction of the force is opposite.

There is some ''fringing'' of the field at the edges where the field is not uniform and the field lines are bent. The electric field is also outside the capacitor, but it is much weaker there. In a number of problems (but not always!) this can be neglected. One should remember that the electrostatic field of a capacitor is potential. It means that the electric field does no work on a charge in the round-trip displacement.


Direct Current Circuits

Direct current circuits consist of batteries, resistors, and meters (ammeters and voltmeters).

The batteries are characterized by the value of electromotive force (emf) [image] and their internal resistance r.

Ohm's law in a part of circuit with emf and a resistor R is as follows:

Complex circuits with junctions are calculated using Kirchhoff's rules.

Junction rule: The algebraic sum of the currents into any junction is zero:

Loop rule: The algebraic sum of the potential drops on the resistive elements in any closed loop must equal the algebraic sum of emfs

Kirchhoff's rules allow one to put down a set of equations enough to determine the currents and potential differences for any circuit.


DC Circuits with Capacitors

A capacitor consists of two oppositely charged conductors separated by an insulator. The form of the conductors is usually chosen so that the electric field is localized in some region (parallel-plate, spherical, cylindrical capacitors).

The capacitance of a capacitor is the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between the conductors

.

The capacitance of a parallel-plate capacitor consisting of the two parallel conducting plates, each  with area S, separated by a distance d, is

,

where [image] is the dielectric constant of the insulator between the plates, [image] [image] is electric constant, S is the area of each plate, and d is the distance between them.

The equivalent capacitance of a parallel combination equals the sum of the individual capacitances

The reciprocal of the equivalent capacitance of a serial combination equals the sum of reciprocals of the individual capacitances

A charged capacitor has the potential energy

.

In the circuit with a serial connection of an emf [image], an ammeter, a resistor R, and a capacitor C, when the connection is made, the current depends on the time as follows

The product [image] is called the relaxation time. After a time equals [image] the current has decreased to the reciprocal of e=2,7.


Magnetic Field of a Straight Conductor

Magnetic field is produced by moving charges (a current).

To characterize the force that magnetic field exerts on a current-carrying conductor one introduces a vector quantity - the magnetic field [image].

The magnetic field [image] can be determined by the magnetic force which acts on a straight conductor with length l carrying the current [image]. If the conductor is at an angle [image] to the vector [image], it is acted upon by the force of magnitude

.

The direction of the force [image] is connected with the direction of the current and the field [image] through right-hand rule.

Magnetic field lines are drawn so that the line through any point is tangent to the magnetic field vector [image] at that point.

The magnetic field lines of a straight conductor are circles around the conductor lying in planes perpendicular to it. In SI the magnetic field at a distance r from a straight conductor carrying the current [image] is expressed as follows

,

where [image] [image] is magnetic constant.

So the magnetic field of the straight conductor is inversely proportional to the first power of the distance [image] from the conductor.


Magnetic Field of a Circular Loop

The magnetic field is produced by moving charges (a current). The magnetic field [image] characterizes the forces the field exerts on a conductor carrying a current.

If a conductor of length l carrying the current [image] is at an angle [image] with [image], it is acted upon by a total force F with the magnitude

.

The direction of the force [image] is connected with the directions of the current and the field [image] through right-hand rule.

The magnetic field can be visualized by magnetic field lines. One draws the lines so that the line through any point is tangent to the magnetic field vector [image] at that point.

The direction of the magnetic field on the axes of a circular loop is along the axes. In the center of the loop with radius R, carrying a current [image], the magnetic field equals

,

where [image] [image] is magnetic constant.

The magnetic field along the axes of the circular loop has a complex behavior as a function of x. When the distance is much larger than the radius of the loop, the field magnitude decreases approximately as [image].


Magnetic Field of a Solenoid

A solenoid consists of a helical winding of wire on a cylinder, usually circular in cross section. The magnetic field [image] far from the ends is uniform and equals

,

where [image] [image] is magnetic constant, [image] is the current in the wire, N is the number of turns, and l is the length of the solenoid.

The direction of the magnetic field is connected with the direction of the current through right-hand rule.

Near the ends of the solenoid the magnetic field is no longer uniform. It decreases and the magnetic field lines are diffused outside the solenoid.


Interaction of Two Parallel Currents

If two parallel conductors carrying currents [image] and [image] are separated by a distance r, the magnetic interaction between them equals

,

where [image] is the magnitude of the force per unit length acting on each of the conductors. If the currents are in the same direction, the conductors attract each other. If the currents are in the opposite direction, they repel each other.

The last formula is the basis of the official SI definition of the fourth basic unit - the ampere.

One ampere is that unvarying current which, if present in each of two parallel conductors of infinite length and one meter apart in empty space, causes each conductor to experience a force of [image] newtons per meter of length.

The magnetic forces between conductors obey Newton's third law:


Motional Electromotive Force

Faraday's law of electromagnetic induction was discovered in 1831.

The induced emf in a circuit equals the negative of the time rate of change of magnetic flux through the circuit:

.

The magnetic flux through the circuit can change due to the two reasons.

1. The magnetic flux changes due to motion of the circuit in a time-independent magnetic field. This is the case of conductors moving in the magnetic field. The appearing of induced emf is explained by the magnetic force acting on the electrons in the moving conductors.

As an example let's consider an induced emf in a rectangular circuit in a uniform magnetic field [image] perpendicular to the plane of the circuit. Let one of the sides with length [image] move with velocity [image] along other two sides. The free charges in this part of the circuit are acted upon by the magnetic force. One of the components of this force, which is connected with the velocity [image] of the conductor, is along the conductor. This component equals

.

The work of the component on a displacement [image] equals

.

By the definition of emf,

.

In the other stationary parts of the circuit emf equals zero. The last equation can be put in a more convenient form. During the time [image] the area of the circuit changes by [image]. The change of the magnetic flux during this time equals [image]. So

.

If the resistance of the circuit is [image], there will flow the current [image].

2. The second reason for the magnetic flux through the circuit to change, is changing of the magnetic field as a function of time, while the circuit remains stationary. In this case appearing of the induced emf cannot be explained by the magnetic force. Electrons in stationary conductor can be moved just by an electric field, and this field is produced directly by the changing magnetic field. This electric field is non-electrostatic. The work of a non-electrostatic electric field done on a unit positive charge, when it goes once around the loop, equals the induced emf in the stationary conductor.

The notion of a non-electrostatic electric field produced by a changing magnetic field was introduced in physics by a great English physicist Maxwell.


Motion of a Charged Particle in Electric Field

The particle with the charge q in the electric field [image] is acted upon by a force

In a uniform electric field a charged particle moves in a parabola like the motion of a particle in gravitational field near the Earth's surface.


Motion of a Charged Particle in Magnetic Field

The magnetic force on a charge q moving with velocity [image] in a magnetic field is given by

where [image] is the angle between the vectors [image] and [image].

The direction of the magnetic force is always perpendicular to the velocity of the particle. The force can change only the direction of the particle's velocity, not its magnitude.

The magnetic force can't do work on the particle.

If the direction of the particle's velocity [image] in a uniform magnetic field is perpendicular to the vector [image], the particle moves with the constant speed and its trajectory is a circle of the radius

If the direction of the initial velocity is not perpendicular to the field [image], the velocity component parallel to the field is constant and the particle moves in a helix. At the same time, in the plane perpendicular to the field, the particle's trajectory is a circle.

The period of one cycle is

and is independent from the speed.


Velocity Selector

In a number of devices, for example, in mass spectrometers it is required to select beforehand the particles with particular velocities. So called velocity selectors are used for this purpose.

In the most simple example charged particles travel in crossed electric and magnetic fields. The electric field is produced between the plates of a parallel-plate capacitor, and the magnetic field is between the poles of an electromagnet. The initial velocity [image] of the charged particles is perpendicular to the vectors [image] and [image].

The particle moving in the crossed electric and magnetic fields is acted upon by the electric force [image] and the magnetic force [image], where [image] is an angle between the vectors [image] and [image].

At some conditions, these forces can be made equal in magnitude, the total force being zero. In this case the charged particle will travel inside the capacitor in a straight line with constant velocity.

Having passed through the capacitor, this particle will then go through a small hole in a screen. The condition of the rectilinear motion is the following:

.

This condition doesn't depend on the charge and the mass of the particle, it depends only on its speed. Given the electric and magnetic fields, the selector filters the particles travelling with the speed

.


Mass Spectrometer

A mass spectrometer is a device which allows one to measure masses of charged particles, for example, those of different nuclei. Mass spectrometers are used to separate isotopes, i.e. the nuclei with the same charges but different masses (for example, [image] and [image]).

A mass spectrometer consists of a chamber with high vacuum. The chamber is in a uniform magnetic field in the region between the two poles of an electromagnet. Charged particles move into the chamber having passed through a velocity selector. The charges travel in the chamber in the plane perpendicular to the magnetic field, as they are acted upon by the magnetic force. Their trajectories are circles of radius

,

where m and q are mass and charge of a particle, v is its speed, and [image] is the magnetic field. Measuring the radius of the trajectory with given v and [image], one can find the ratio [image]. In the case of isotopes [image] the mass spectrometer allows one to separate particles with different masses.


Faraday Experiments II
Faraday Experiments I

Faraday experiments on electromagnetic induction were carried out during the 1830s.

The phenomenon of electromagnetic induction is appearing of a current in a circuit (coil) when magnetic flux through the circuit changes.

The magnetic flux [image] through the area [image] is

,

where [image] is the magnitude of the magnetic field, [image] is an angle between the vector [image] and the normal [image] to the plane of  the circuit.

In the case of a long coil the net magnetic flux is the sum of individual magnetic fluxes through each turn of the coil.

The magnetic flux through a coil can be changed due to two reasons:

- change of the magnetic field with time

- motion of the coil in the magnetic field.

In both cases the electromagnetic induction goes similarly.

Faraday's law expresses the induced emf in a circuit through the rate of change of the magnetic flux

.

The direction of the induced current can be determined using Lenz's law: the direction of any magnetic induction effect is such as to oppose the cause producing it.

The source of the magnetic field producing the magnetic flux through the coil can be a magnet or another coil with a current.

The induced current appears when there is a relative motion of the coil and the source of the magnetic field. If the magnetic field is produced by the other coil, the induced current emerges also when the switch opens or closes the second coil.


Faraday Experiments II

Faraday experiments on electromagnetic induction were carried out during the 1830s.

The phenomenon of electromagnetic induction is appearing of a current in a circuit (coil) when magnetic flux through the circuit changes.

The magnetic flux [image] through the area [image] is

,

where [image] is the magnitude of the magnetic field, [image] is an angle between the vector [image] and the normal [image] to the plane of  the circuit.

In the case of a long coil the net magnetic flux is the sum of individual magnetic fluxes through each turn of the coil.

The magnetic flux through a coil can be changed due to two reasons:

- change of the magnetic field with time

- motion of the coil in the magnetic field.

In both cases the electromagnetic induction goes similarly.

Faraday's law expresses the induced emf in a circuit through the rate of change of the magnetic flux

.

The direction of the induced current can be determined using Lenz's law: the direction of any magnetic induction effect is such as to oppose the cause producing it.

The source of the magnetic field producing the magnetic flux through the coil can be a magnet or another coil with a current.

The induced current appears when there is a relative motion of the coil and the source of the magnetic field. If the magnetic field is produced by the other coil, the induced current emerges also when the switch opens or closes the second coil.


The Alternating Current Generator

According to Faraday's law of electromagnetic induction, when a magnetic flux [image] through a circuit (coil) changes, there appears an induced emf depending on the rate of change of magnetic flux:

.

The magnetic flux [image] through the area [image] equals, by definition,

,

where [image] is a magnitude of the magnetic field, and [image] is an angle between [image] and the normal to the plane of the circuit.

If a rectangular frame with area [image] rotates at uniform angular velocity making f turns per second in a uniform magnetic field [image], the magnetic flux through the frame is a periodic function of time:

.

So between the ends of the frame there appears a periodic voltage which equals

.

The amplitude of this voltage is proportional to the angular velocity of the frame. Such a frame is a model of an alternating current generator. If the terminals of the frame are connected to a special device (collector), one can get a direct current generator.


Torque on a Current-Carrying Loop

A loop with the current I in a uniform magnetic field [image] is acted upon by a torque [image]. It is the result of magnetic forces acting on every element of the loop. The net torque equals

.

Here [image] is the area of the loop, [image] is an angle between a normal to the plane of the loop and the direction of the magnetic field [image]. The product [image], where [image] is the unit vector normal to the plane, is called the magnetic moment of the loop. The direction of the magnetic moment [image] is connected with the direction of the current in the loop through right-hand rule.

The torque [image] tends to turn the loop so that the direction of the magnetic moment [image] coincides with the direction of the magnetic field [image]. When they coincide, the loop is in stable equilibrium position [image]. If [image]  and [image] are antiparallel, the equilibrium is unstable.

Most easily the torque can be found in the case of a rectangular current loop, assuming that [image] and the axes of the turn coincides with the z-axes. In this case the torque is the result of a couple of forces acting on the loop's sides parallel to the axes of the turn.


The R-C Circuits

If a capacitor with capacitance [image] charged initially by the charge [image] is connected to a resistor of resistance [image], it starts to discharge, and this process is described by the formula

,

where [image] is the time constant (or the relaxation time) of the [image] circuit. In a time [image] the charge of the capacitor will decrease by a factor of e=2,7. In the process of discharging the energy stored in the capacitor is released on the resistor as a heat.


The R-L Circuits

If a coil with inductance [image] is connected in series with a resistor of resistance R to a source with emf V, the current in the circuit will increase as follows:

,

where [image] is the time constant (relaxation time) of the [image] circuit. The maximum value of the current [image] will be reached in a time much larger than [image]. In a time [image] the current I will constitute [image] of its maximum value. The reason for such smooth change of the current is the self-induced emf in the coil.


Free Oscillations in L-R-C Circuits

The processes in the circuit after it is forced out of equilibrium and is let free after that, are called free processes.

In a circuit with a capacitor [image], an inductor [image] and a resistor [image], these processes can be of oscillating character. An initial disturbance can be a charge on the capacitor [image], given before the connection is made.

In an ideal circuit without losses [image], the following process is that of free undamped oscillations at the natural frequency of the circuit

.

This process can last infinitely long. Two times during a period the energy stored in the capacitor transfers to the magnetic energy of the inductor.

With the energy losses ( [image]), there occur damped oscillations in the circuit. The amplitude of the oscillations decreases exponentially with time. The time after which the amplitude has decreased by a factor of e=2.7 is called the relaxation time. It equals [image].

If the resistance [image] is large enough, the relaxation time becomes comparable with the period [image]. In such a case the free process in the circuit is no longer an oscillating one.

In real circuits there are always energy losses. In the end all the electric energy stored initially in the capacitor transfers to heat released on the resistor.


Forced Oscillations in L-R-C Circuits

The settled oscillations in a circuit, when a harmonic voltage [image] from a generator is present, are called forced oscillations. They are always of the same frequency [image] as that of the generator.

If the external harmonic voltage is in the [image] circuit, the amplitude of the forced oscillations (for example, the amplitude of the voltage on the capacitor) depends very much on the relationship between the frequency [image] of the generator and the natural frequency [image] of the circuit. At [image] occurs a resonance - the ratio of the amplitude of the forced oscillations to the amplitude of the external generator reaches its maximum. The graphic dependence of this ratio as a function of the frequency of the generator is called the resonance curve. When the resistance of the circuit increases, i.e. increases the loss of energy, the resonance curve becomes less sharp.

There is a phase shift between the source voltage and the voltage on the capacitor depending on the relationship between [image] and [image]. At the resonance the shift is [image].

The relationships between the amplitudes of the voltages and currents are conveniently analyzed with the use of vector (phasor) diagrams. This is a graphic representation of harmonic oscillations by vectors (phasors). Two (or several) harmonic oscillations with the same generator's  frequency [image] are represented on the diagram by phasors. Their lengths equal the amplitudes of the oscillations and the angles between them equal the phase shift. So the phasor representing the voltage of the coil should be turned [image] counter-clockwise with respect to the phasor representing the current in the coil. It is said that the voltage of the coil leads the current by [image]. On the contrary, in the capacitor the current leads the voltage by [image]. In the circuit, the vector sum of all the voltages of elements connected in series must equal the phasor representing the external generator's voltage.

In the L-R-C series circuit at the resonance, the amplitudes of the voltages [image] and [image] of the capacitor and the coil respectively are the same, and the phase shift between them is [image]. That's why the phasor of the external voltage coincides with the phasor of the resistor.


Reflection and Refraction

When a light wave strikes a smooth border between two transparent media, the wave can be partly reflected and partly refracted into the second material. In geometric optics the laws of reflection and refraction are based on a notion of rays.

The law of reflection: the incident and reflected rays, and the normal to the surface all lie in the same plane. The angle of reflection [image] is equal to the angle of incidence [image].

The law of refraction: the incident and refracted rays, and the normal to the surface all lie in the same plane. The ratio of the sines of the angles of incidence [image] and refraction [image], where both angles are measured from the normal to the surface, is equal to a constant for the two given materials:

.

The constant n is called the relative index of refraction of the second material with respect to the first. The index of refraction of an optical material with respect to vacuum is called the index of refraction.

The relative index of refraction of two optical materials equals the ratio of their indexes of refraction

.

When a ray is incident on an interface with a second material whose index of refraction is smaller than that of the material in which the ray is traveling [image] (for example, from glass to air), one can see total internal reflection, i.e. disappearance of the refracted ray. This can be seen at angles of incidence beyond the critical angle [image].

For the incident ray [image]  [image], [image].

If the second material is air ( [image]) the formula can be put in a more convenient form

,

where [image] is the index of refraction of the first material.


The Plane Mirror

When rays are reflected by a plane mirror, there appears a virtual image. The object and its virtual image are symmetric about the mirror surface. Such a symmetry in position of the object and its image is a consequence of the law of reflection: the incident and reflected rays, and the normal to the surface all lie in the same plane. The angle of reflection is equal to the angle of incidence.

The image is the same size as the object.


The Spherical Mirror

A spherical mirror is a reflecting mirror in a form of a spherical segment. The center of the sphere of which the surface is a part is called the center of curvature. The point at the center of the mirror surface is called the vertex of the mirror. The line passing through the center of curvature and the vertex of the mirror is called the optic axis.

Spherical mirrors can be convex and concave.

If the incoming rays are parallel to the optic axis, they converge at a point which is called the focal point. The distance between the focal point and the vertex of the mirror is called the focal length. A concave mirror has a real focal point. It is in the middle between the center and the vertex.

The focal point of a convex mirror is a virtual one. In this case the incoming rays parallel to the optic axis are not reflected through the focal point. Instead, they diverge as though they had come from the focal point. The focal length is ascribed a particular sign: the concave mirror has f>0, and the convex one has f<0. In both cases the magnitude of the focal length is one half of radius of curvature: .

Spherical mirrors, like lenses can make images of objects. The images can be erect and inverted, real and virtual. The position of the image can be determined by graphical methods. To do this one can use properties of the rays passing through the center of the mirror or its focal point, as well as the rays parallel to the axis.

The position of the image can also be found by the formula similar to that of a thin lens:

.

The rules for determining signs of [image] and [image] are the same as for a thin lens:

 and [image] for real objects and images;

 and [image] for virtual objects and images.

The lateral magnification of the spherical mirror: . Here [image] and [image] are sizes of the object and its image.

The signs in this formula are such that for an erect image m>0; and for an inverted m<0.


The Lens

A lens is an optical system with two refracting surfaces. The simplest lens has two spherical surfaces close enough together that we can neglect the distance between them (the thickness of the lens). Such a lens is called thin.

The main quality of the lens is its ability to produce images of objects. The images can be erect and inverted, real and virtual.

The position of the image and its type can be found by graphical methods. To do this one can use properties of the rays passing through the center of the lens or its focal points, as well as the rays parallel to the optic axis.

The image can also be found from the thin lens equation. If the distance between an object and the lens is denoted as [image] and the distance between the lens and the image is [image], the thin lens equation is given by

.

The quantity f is called the focal length. It equals the distance between the center of the lens and the focal point, i.e. the point where the rays parallel to the optic axis converge. Lenses can be converging and diverging. The converging lens has real focal point - upon passing the lens a ray converges to a real point. The diverging lens has virtual focal point.

The focal length is ascribed a particular sign: for a converging lens f>0, while for a diverging f<0.

The distances [image] and [image] are also ascribed particular signs:

 and [image] for real objects and images,

 and [image] for virtual objects and images.

Depending on the position of the object with respect to the lens, the size of the image changes. The ratio of the linear size of the image [image] to the size of the object [image] is called the lateral magnification:

.

The signs in the formula are such that for an erect image m>0, and for an inverted m<0.


The System of Two Lenses

In many optical instruments the light passes through two or more lenses. For example, the simplest optical schemes of a telescope and a microscope include two thin lenses - the objective and the eyepiece (the ocular). The image of an object produced by the first lens is an object (real or virtual) for the second lens which produces the second image of the object. This second image can also be real or virtual. So the computation of the system of two lenses reduces to using the thin lens equation

twice.

The distance [image] from the first image to the second lens should be set equal to [image] where l is the distance between the two lenses. The value of [image] found from the equation determines the position of the second image and its type ( [image] for the real image and [image] for the virtual one).

If the object and its both images are within finite distance from the lenses, one can introduce the lateral magnification for the lenses

, [image].

The total lateral magnification of the system of two lenses equals the product of lateral magnifications of the lenses.

If the object or its images are in the infinity, the lateral magnification of the system of two lenses is senseless.

A particular case is that of a telescope with both the object and its second image in the infinity.


The Eye

The optical behavior of the eye is similar to that of a camera. The light rays from an object refracting at the cornea pass then through the chrystalline lens (a lens with the changeable focal length) and produce an image on the retina.The retina plays the same role as the film in a camera. The main peculiarity of the eye is its ability to change its optical properties depending on the position of an object. This quality is called accommodation.

The extremes of the range of over which distinct vision of the normal eye is possible are known as the far point and the near point of the eye.

At the distance of normal vision it is most convenient (without too much strain) to consider details of an object, for example, to read the text.

The far point of a normal eye is at infinity. The distance of normal vision of normal eye is taken to be [image] cm. The position of the near point depends on the amount by which the ciliary muscle can increase the curvature of the chrystalline lens.

In the myopic (nearsighted) eye rays from an object at infinity are focused in front of the retina. In the hyperopic (farsighted) eye the image of an infinitely distant object is behind the retina.

The distance of normal vision of the normal eye is more than of the myopic eye and less than of the hyperopic eye.

All these defects can be corrected by the use of eyeglasses. The hyperopic eyes need positive (converging) lenses, and the myopic eyes need negative (diverging) lenses.


The Telescope

A telescope is designed to look at distant objects. It consists of two lenses - a lens with a large focal length turned to an object (the objective) and a lens with a small focal length turned to the viewer (the eyepiece). Telescopes can be of two types:

- an astronomical telescope. It gives an inverted image of a distant object and so is inconvenient for looking at earthen objects. The ocular of such telescope is a converging lens.

- a Galilean telescope is designed for looking at earthen objects. Such telescope gives an erect magnified image of a distant object. The ocular in the Galilean telescope is a diverging lens.

When analyzing the way of rays in a telescope, it is usually assumed that the viewer's eye is accommodated at infinity; that's why the rays after the telescope are parallel. This is achieved when the distance between the objective and the eyepiece equals the sum of their focal lengths [image].

When the object (and its image) is in infinity, it is senseless to speak about the lateral magnification. Telescopes are characterized by the angular magnification M which is defined as the ratio of the angle [image] subtended at the eye by the final image to the angle [image] subtended at the unaided eye by the object (for small angles). The angles [image] and [image] are those between the optic axis of the system and the parallel rays from some point of an object or its image. It is convenient to take these angles with the signs plus or minus so that the sign of the angular magnification M represents the type of the image (erect or inverted).

This magnification can be easily expressed by the focal lengths of objective and eyepiece

.


The Microscope

The microscope is used for getting great magnification when looking at small objects. In the microscope the magnified image is produced by an optical system consisting of the objective and the eyepiece (also called the ocular). As an objective there is used a lens with a short focal length. The object is put beside the objective at a distance [image] near the first focus. In this case the objective gives a magnified inverted image which is viewed by the eye through the eyepiece. A converging lens with a short focal length is used as an eyepiece.

When analyzing the way of rays, it is convenient to take that the eye is accommodated at the distance of  normal vision [image] cm. The viewer can look at a small object directly putting it at a distance [image] from the eye, or through the microscope seeing the virtual image of the object produced by the eyepiece from the same distance [image]. The ratio of the angle subtended at the eye by the final image to the angle subtended at the unaided eye by the object is called the angular magnification. As in both cases the object is viewed from the same distance [image], the angular magnification coincides with the lateral magnification of a system of two lenses.

The virtual image of the object viewed through the eyepiece is always inverted. If it is inconvenient (for example,  reading tiny script), the object itself can be inverted. So the magnification of the microscope is always characterized by a positive value.

The magnification M is determined as the product of the magnifications of the objective and of the eyepiece. In the case when [image] and [image] are much smaller than the distance [image] between them, the microscope's magnification is expressed by the formula

.

The standard microscope has [image] mm.

The optical microscope can give the magnification as large as several hundreds. At a very large magnification diffraction phenomena starts to appear.


Newton's Rings

If a convex surface of a lens is in contact with a plane glass plate, a thin film of air is formed between the two surfaces. When you view the setup with monochromatic light, you see circular interference fringes which are called Newton's rings. The radii [image] of Newton's rings depend on the wavelength [image] of the incident light. The center of the pattern is always dark.

This phenomenon was explained by T. Young using the wave concept. Newton's rings are the result of interference between the waves reflected by the surface of the lens and the plane surface of the plate. There appears the path difference between these two waves which equals double thickness of the air film in that place (if the rays of the incident light are normal to the surface). If this path difference equals an integral multiple of the wavelength, both waves reinforce each other and there is a maximum amplitude. If the path difference equals a half-integral  number of wavelengths, both waves cancel each other and there is a minimal amplitude. The phenomenon is complicated by changing the phase of the wave by [image] during the reflection at the interface between air and the glass plate, which corresponds to a [image] increase in the path difference.  This results in the minimal amplitude (dark spot) in the center of the pattern where the thickness of the air film is much less than the wavelength [image].

The calculation shows that the radius [image] of the m'th dark ring is given by

,

where R is the radius of curvature of the convex surface of the lens. By measuring in the experiment the dark ring's radii, one can determine the radius of curvature of the surface of the lens given the wavelength [image] of the light.


Interference Experiment of Young

When two waves overlap, the resultant wave amplitude is greater in some points and smaller in others. This effect is called interference. In interference schemes rays of light come from the source along two different paths. The result of interference depends on the path difference [image] between the two waves and the wavelength [image]. If the path difference equals an integral number of wavelengths [image], there is constructive interference, or reinforcement, (a bright fringe).

Similarly, destructive interference, or cancellation, (a dark fringe) occurs, at points for which the path difference is a half-integral number of wavelengths [image].

The width [image] of the fringes on the screen (the distance between nearest bright or dark fringes) depends on the wavelength [image] and an angle [image] between the interfering rays. If the screen is placed symmetrically with respect to the incident rays, [image].

The historically first experiment, which was explained by the wave theory of the light, was performed by the English scientist T. Young in 1800. In his experiment light from a source passed through two closely situated slits. The light beams broadening due to diffraction fall on a white screen. In the area where the beams intersect there are interference fringes.

If the distance between the double slit and the screen is [image] and the distance between the slits equals d, the angle [image] between the rays falling on the screen is [image] (if [image]) and the width of the fringes is given by

.

By measuring the width [image] T. Young for the first time determined the wavelengths of light of different colors.


Diffraction

Diffraction is bending of light around an obstacle. If parallel rays strike a round obstacle (a disk, a ball or a circular aperture in an opaque screen), on a distant screen there appears a set of alternating bright and dark rings. If the obstacle is of linear form (a slit, a thread, or a screen's edge), there appears a set of parallel diffraction fringes on the screen.

The theory of diffraction phenomena was developed by the french physicist Fresnel in 1818 on the basis of the wave concept. Fresnel complemented Huygens' principle by an idea about interference of secondary wavelets. This allowed to calculate diffraction patterns from different simple obstacles.

The diffraction theory was based on the notion of the Fresnel zones. For round obstacles the radii [image] of the circular Fresnel zones equal

    (m=1, 2, 3 ...)

where [image] is a wavelength, and L is the distance between the obstacle and the screen. The result of diffraction depends very much on the number m of the Fresnel zones on the obstacle ( [image])

.

In the case of the difraction by a circular operture in an opaque screen there is a bright spot in the center of the difraction rings if the number m is odd, if m is even, the spot is dark.

In diffraction on a round disk (or a ball) there is always a bright spot in the center of the pattern (the Poisson bright spot).

In diffraction by a single slit of the width d the character of the diffraction pattern also depends on the number m of the band-like Fresnel zones determined by

.

The number of the Fresnel zones on the obstacle determines the character of the diffraction pattern. This allows one to simulate diffraction phenomena using waves of a different spectrum band and choosing appropriate geometric size of the apparatus. For example, the diffraction of radiowaves can be simulated by light.


Fresnel's Zones

To explain basic laws of light propagation near obstacles (apertures, slits, opaque disks) Fresnel complemented Huygens' principle by a postulate about interference of secondary wavelets coming to the viewed point from different parts of the initial wave front unshaded by opaque obstacles. This new postulate was called Huygens-Fresnel principle.

This principle allowed to solve a number of diffraction problems and calculate diffraction patterns from different obstacles. One of the simple problems is that of diffraction on a circular aperture or a disk. If a point source and the viewed point are on the axis of the system, the wave front is convenient to divide into the circular Fresnel zones.

The radii [image] of the Fresnel zones in the screen's plane are given by [image]   (m = 1, 2, 3 ...)

where [image] is a wave length, and [image] is the distance between the obstacle and the screen.

The calculations show that areas of the Fresnel zones are equal. So are equal the amplitudes of secondary wavelets in the viewed point. But the phases of oscillations in the point P excited by neighboring zones differ by [image].

That's why, when increasing the size of the aperture, the light intensity will be zero if there is an odd number of zones on the aperture, and will reach its maximum if there is an even number.

If one shades only odd or only even Fresnel zones, he can get a large increase of intensity in the point P, as such oscillations of only odd or only even are in phase. Such devices are called the Fresnel zone plates. They act like a lens.

In diffraction on a round disk there is always a bright spot in the center of the diffraction pattern (the Poisson bright spot). This result can also be obtained from the theory of the Fresnel zones.

A general diffraction pattern in the diffraction on a round obstacle is rather complex. It can be found for each viewed point by adding oscillations from secondary sources. The calculation of the result of interference of the secondary wavelets is a complex mathematical problem. It can be solved by the use of computer simulation.


The Diffraction Grating

The simplest diffraction grating is an array of a large number of parallel slits, all with the same width and spaced equal distances between centers.

If a grating is illuminated by a parallel beam of light, after the grating the light travels in several directions. If one puts a converging lens behind the grating, in its focal plane there will be diffraction maxima of different orders. These maxima are called principal. The beams of light after the grating pass in the directions given by

Here d is the period of the grating, [image] is the wavelength, and m is an integer called the order of diffraction maximum.

At small angles the distance between  the zero-order maximum (m=0) and the m'th order maximum in the focal plane of the lens of focal length f is

.

As the position of a maximum (except for the zero one) depends on the wavelength, the grating can disperse a light beam into spectra, i.e. it is a spectral device.

With the use of a diffraction grating one can carry out very precise measurements of the wavelength. Given the grating period, the measurement of the wavelength reduces to the measurement of the angle [image] corresponding to the direction of the m'th maximum.

If the light consists of two monochromatic waves of wavelengths [image] and [image], the grating can separate one from the other in every order (except for m=0).


Circular Apertures and Resolving Power

If a parallel beam of light passes through an aperture in an opaque screen, it broadens due to diffraction and is no more parallel. In the focal plane of a lens placed behind the screen the image of the source is diffused due to diffraction. In the case of a circular aperture the diffraction image consists of the central bright spot (the so called Airy disk) and a number of concentric circles around it.

The radius of the central spot in the focal plane of the lens is

where [image] is the wavelength, [image] is the diameter of the aperture, and f is the focal length.

If the rays from a distant source fall on a lens directly, the aperture of the lens plays the role of the screen. In this case [image] is the diameter of the lens.

The size of the diffraction spot increases with the increase of the wavelength [image] and with the decrease of the lens's diameter D.

The diffraction character of the image limits abilities of optical instruments. For example, the diffraction images of two close stars may strongly overlap so that one cannot distinguish whether there is one star or two.

If the light from two stars at an angular separation [image] strikes the objective of a telescope, in the focal plane of the objective lens of diameter [image] with the focal length f, the centers of diffraction maxima will be [image] apart (for small angles [image]). If this distance equals the radius [image] of the central spot, the diffraction patterns strongly overlap so that one cannot distinguish visually the image of two stars from the image of one star. According to this criterion (called Rayleigh's criterion), the quantity

is called the difraction limit of resolution of a lens.

In a similar way the wave nature of light limits ability to resolve close details of an object viewed through a microscope.


Polarizers

Light has electromagnetic nature. The electromagnetic waves are transverse: the vectors [image] and [image] are perpendicular to each other and the oscillations are in the plane perpendicular to the direction of propagation.

The light waves emitted by usual sources of light (for example, by a lamp) are unpolarized. This means that the oscillations of the vectors [image] and [image] are in all possible transverse directions. Such light is called natural.

Some of the sources (lasers) can emit polarized light. In this light the oscillations of the electric and magnetic fields are not in all directions, but only in the two perpendicular ones. Such light is called linearly polarized.

There are different optical devices able to turn unpolarized light into polarized one (for example, a tourmaline crystal). So-called polaroid have the same ability. A polaroid is a thin film of hepatite crystals. After passing through polaroid unpolarized light becomes linearly polarized. The direction of the electric-field vector in the transmitted wave is called the polarizing axis.

Polaroids are used for obtaining polarized light and analyzing it (polarizers and analyzers).

If natural light passes through two consequent polaroids, the intensity of the transmitted light depends on an angle [image] between the polarizing axes of the polaroids:

.

This relation is called Malus's law.


Polarization of Light

Light has electromagnetic nature. The electromagnetic waves are transverse: the vectors [image] and [image] are perpendicular to each other and the oscillations are in the plane perpendicular to the direction of propagation.

The light waves emitted by usual sources of light (for example, by a lamp) are unpolarized. This means that the oscillations of the vectors [image] and [image] are in all possible transverse directions. Such light is called natural.

Some of the sources (lasers) can emit polarized light. In this light the oscillations of the electric and magnetic fields are not in all directions, but only in the two perpendicular ones. Such light is called linearly polarized.

A general case is elliptical polarization of light. The elliptically polarized light can be presented as a superposition of two waves of the same frequency linearly polarized in mutually perpendicular planes. The type of elliptical polarization depends on the relationship between the amplitudes [image] and [image] of linearly polarized waves and a phase shift [image] between them. In particular, if [image],

for [image] or [image] the polarization is linear;

for [image] or [image] the polarization is circular (left or right).

In any plane perpendicular to the direction of light propagation, the end of the vector [image] in an elliptically polarized wave moves periodically along an elliptic trajectory.


The Photoelectric Effect

The photoelectric effect is the emission of electrons from surface of a conductor when light strikes its surface. The photoelectric effect was first observed in 1887 by Hertz. The correct analysis of the photoelectric effect was developed by Albert Einstein in 1905 on the basis of quantum notions. The classical wave theory of light couldn't explain this effect.

Einstein postulated that a beam of light consists of small bundles of energy called quanta or photons. The energy [image] of a quantum is proportional to its frequency

where [image] J s is a universal constant called Planck's constant.

If an electron gets energy which is greater than work function A, it can escape from the surface. According to Einstein, the maximum kinetic energy for an emitted electron is determined by the following equation

.

This equation explains basic features of the photoelectric effect:

1. Number of electrons emitted from the surface per second is proportional to the radiation flux.

2. The maximum kinetic energy of photoelectrons linearly increases with the frequency [image] and doesn't depend on the incident light flux.

If between the cathode and the anode of a phototube there is an electric field pushing electrons back to the anode, the electron flow stops when the reversed potential reaches the stopping potential [image] given by

.

No photoelectrons are emitted unless the frequency of the light is greater than some minimum value called the threshold frequency

, or [image].

Alkaline metals have their threshold frequency in the region of visible light.


The Laser. Two Levels Model.

The name laser is an acronym for light amplification by stimulated emission of radiation. One can understand the principles of laser operation on the basis of photons and atomic energy levels.

If a quantum system with two levels interacts with radiation, there can occur different processes.

1. Absorption of a photon. In this case an atom goes from a lower to an upper (excited) state.

2. Spontaneous emission when the excited atom returns to the lower state by emitting a photon. The direction and phase of the emitted photon are random, so the spontaneous emission is incoherent.

The frequency of the absorbed and emitted photons is determined by the energy difference between the two levels:

.

3. Stimulated emission from the excited to the lower state. This process predicted by A. Einstein in 1916 lays in the basis of laser physics. In stimulated emission an incident photon encounters an excited atom. By a kind of resonance effect, the photon forces the atom to emit another photon with the same frequency, the same direction, the same phase, and the same polarization as the incident photon.

All the three processes occur simultaneously when an external resonant beam of light passes through the two - levels system. The intensity of this beam even decreases as some excited-state atoms lose their energy and come to the lower state without emitting a photon (for example, due to inelastic collisions with other atoms). For the passing through the medium beam to increase, one has to enhance the number of atoms in excited states by some external source of energy. This can be achieved in a variety of ways, but all these mechanisms are impossible in a two-level system. The supply of energy must be done via a third higher level. This process must create a non-equilibrium situation in which the number of excited atoms is larger than the number of lower-state atoms. Such a situation is called a population inversion.

When the number of excited atoms is large enough, the rate of energy radiation by stimulated emission exceeds the rate of absorption. The system then acts as a source of radiation. For a laser to work like this, the system should be put between two parallel mirrors with high index of reflection. Then the increasing light passes after many reflections through the medium causing an avalanche of emitted photons.

So emerges the laser emission.


The Bohr's Pastulates
The Bohr's Postulates

The first step for developing the quantum theory of atoms was made by Niels Bohr in 1913. He stated his ideas in a form of postulates.

1. Each atom has a set of possible energy levels (stable states). An atom can have an amount of internal energy equal to any of these levels, but it cannot have an energy intermediate between two levels. In stable states an atom does not radiate energy.

2. An atom can make a transition from one energy level to another by emitting a quantum of electro-magnetic energy (photon). The energy of an emitted or absorbed photon equals the energy difference between the levels:

where [image] J s is Planck's constant, and [image] is the photon's frequency.

Both these postulates contradict laws of classical physics, but they well agree with experimental data.

3. The third Bohr's postulate assumes that an electron in an atom can revolve in certain circular stable orbits with the angular momentum being quantized:

where m is the electron's mass, v is its speed, and [image] is the orbit radius. The integer n is called the quantum number.

Applying Bohr's quantum postulates to circular orbits of the hydrogen atom results in the following equations:

The radii of the orbits are:

where [image] m is the Bohr radius.

The total energies of the states are:

.

The state with the least energy (n=1) is called the ground state. For the hydrogen atom [image] J [image] eV. This energy is called the ionization energy.


Relativity of Length

Einstein's special theory of relativity is based on just two simple postulates which are a generalization of experimental data:

1. The principle of relativity: The laws of physics are the same in every inertial frame of reference.

2. The Principle of constancy of the speed of light: The speed of light is the same in all inertial frames of reference and is independent of the motion of the source.

One of important consequences of the special theory of relativity is the conclusion about relativity of length.

Length is not absolute, but depends on body's velocity relative to a given frame of reference.

Let the length of a ruler in the rest frame (where it is at rest) be [image]. The length [image] of the ruler in another frame of reference, moving with a velocity [image], is the distance between the ends of the ruler fixed simultaneously by the watches of this frame. Then, according to the theory of relativity, the following equation is valid:

where [image].

So the length of a moving ruler is always shorter than the length measured in the rest frame.

The relativity of length is connected with the speed of light being the same in all inertial frames of reference and relativity of time intervals.


The Electron Diffraction

In the basis of modern quantum physics there is a notion about duality of all the material objects: they have both wave and particle properties. According to de Broglie's hypothesis made in 1924, every body with mass m, moving with speed v, should have a wavelength

where [image] J s is Planck's constant.

The wave aspects are most pronounced in microobjects (fundamental particles). Due to small mass the de Broglie wavelength appears to be comparable with distances between adjacent atoms in crystals. In this case there are diffraction phenomena if a beam of particles interacts with the crystal lattice. Electrons with the energy 150 eV have a wavelength [image] m. The distances between atoms in crystals are of the same order. If a beam of such electrons is directed towards a crystal, they will be scattered according to diffraction laws. The diffraction pattern fixed on a film contains information about the structure of the three-dimensional crystal lattice.

Electron scattering in crystals can be illustrated by the mental experiment of electron diffraction on the one - dimensional grating. From the wave point of view, this model is completely equivalent to the optical experiment with the diffraction grating. The positions of maxima are determined by

where [image] is the grating spacing, [image] is the angle of diffraction, [image] is an integer (the order of diffraction maximum), and [image] is the de Broglie wavelength.

At small angles of diffraction

.

If a photographic film is placed at a distance [image] from the grating, there will be a diffraction pattern with a number of thin fringes whose positions for small angles of diffraction are determined by

.

In quantum physics the intensity distribution in the diffraction pattern is interpreted as a distribution of probability of an electron striking different points of the screen. Each electron interacts with the grating as a wave (so it interacts with the whole grating), but on the screen it is localized in a definite point. So the diffraction pattern appears as a result of the probability process.


The Compton Scattering

A phenomenon called Compton scattering was first observed in 1924 by A. H. Compton. When x rays strike matter, some of the radiation is scattered on the electrons of light atoms. Compton discovered that some of the scattered radiation has longer wavelength than the incident radiation and that the change in wave length depends on the angle through which the radiation is scattered. Compton scattering is explained with the notion of photons of energy [image] and momentum [image] where [image] J s is Planck's constant, and [image] is the photon's frequency. The process of scattering reduces to collisions between the photons and electrons who can be approximately considered as free.

A calculation based on laws of conservation of energy and momentum gives the following equation:

where [image] is the wavelength of the incident radiation, [image] is the wavelength of the scattered radiation, [image] is the mass of electron, c is the speed of light, and [image] is an angle of scattering.

The quantity

 nm

is called the Compton wavelength.

In the spectrum of the scattered radiation along with the longer-wavelength peak of the wavelength [image] there is also a shorter-wavelength peak at the same wavelength [image] as that for the incident x rays. It corresponds to x-ray scattering from tightly bound electrons. The relationship between the intensities of the peaks depends on the material.

Compton scattering is a bright confirmation of quantum theory.


The Binding Energy of Nucleus

All the nuclei consist of protons and neutrons. The number of protons [image] is equal to the number of an element in the periodic table of the elements. The number of the neutrons is designated by [image]. The total number of protons and neutrons

is called mass number.

The nuclei with the same number of protons and different number of neutrons are isotopes of an element. The general format for an element X with mass number A and the number of protons [image] is

, or [image].

The forces that hold protons and neutrons together in the nucleus are called nuclear forces.

A very important concept in nuclear physics is that of the binding energy. The binding energy is the energy required for total fission of the nucleus into separate particles. It equals the energy which is released when the nucleus is formed from separate particles. The binding energy is determined on the basis of Einstein's relation between mass and energy

.

Exact measurements of nuclei's masses show that the total mass of a nucleus M is always less than the total mass of its constituent parts. The quantity

is called the mass defect. The binding energy equals

where c is the speed of light.

The ratio of the binding energy [image] of a nucleus to the number of nucleons A is called binding energy per nucleon

 (MeV/nucleon).

Binding energy per nucleon is different for different nuclei. The dependency of [image] as a function of the mass number A is investigated experimentally for all stable nuclides. Except for the lightest nuclides, binding energies per nucleon are about 8 (MeV/nucleon). This  is almost a million times larger than the ionization energy of the hydrogen atom.

The nuclei of stable isotopes are formed only with the definite relationship between the numbers of protons and neutrons. The dots representing stable isotopes on the plane [image] are grouped in the limits of a quite thin band. For light nuclei the number of neutrons is approximately equal to the number of protons, but with increasing mass number, the growth of [image] leaves behind the growth of [image].


The Wave Properties of Particles

Light behaves as a wave and particle at the same time. Its properties as a wave show themselves during propagation of light (interference, diffraction). The properties as a particle are significant during interaction of the light with the matter (the photoelectric effect, emission and absorption of light by atoms).

The properties of a photon as a particle (energy E and momentum p) are connected with its properties as a wave (frequency [image] and wavelength [image]) by the equations

; [image],

where [image] J s is Planck's constant.

According to de Broglie's hypothesis made in 1924, every body with mass m, moving with speed v, should have a wavelength

.

The wave aspects are most pronounced in fundamental particles. Due to small mass, the de Broglie wavelength appears to be comparable with distances between adjacent atoms in crystals. In this case there is diffraction if a beam of particles interacts with the crystal lattice. For example, electrons with the energy 150 eV have a wavelength [image] m. The distances between atoms in crystals are of the same order. That's why a beam of electrons will be scattered on a crystal like a wave, i.e. according to diffraction laws.

To illustrate the wave nature of particles there is often used a model with a beam of electrons (or other particles) passing through a slit of width [image]. From the wave theory's point of view, after diffraction on the slit, the beam broadens with an angular divergence [image]. From the particle theory's point of view, the broadening of the beam after passing through the slit is explained by appearing of a certain transverse momentum of a particle. The uncertainty of this transverse momentum is

.

The relation

is so called Heisenberg uncertainty principle. This uncertainty reflects the properties of a particle as a wave in terms of its properties as a particle.

The experiment of an electron beam passing through two slits may be an even more brighter illustration of the wave properties of a particle. This experiment is an analog of the two-slit interference experiment by Young.


Relativity of Time

Relativity of time is one of important consequences of Einstein's special theory of relativity.

When two observers moving relative to each other measure a time interval, they may not get the same result.

If two events occur at the same point in space in a particular frame of reference, and if the time interval between them, as measured by an observer at rest in this frame, is [image], then an observer in a second frame moving with constant velocity u relative to the first frame will measure the time interval to be [image], where

.

Here c is the speed of light.

The interval [image] is always larger than the interval [image]. This effect is called time dilation. This conclusion of the theory of relativity is a consequence of the postulate about the speed of light being the same in different frames of reference.

The time interval [image] measured between two events that occur at the same point in space is called the proper time.


Standing Electron Waves on a Circular Orbit
Standing Electron Waves on Circular Orbits

A very important role in developing modern understanding of atom's structure was played by the Bohr model. In 1913 the Danish physicist N. Bohr stated the following postulates:

1. Each atom has a set of possible energy levels (or quantum states) with a definite energy [image] associated with each state. An atom can have an amount of internal energy equal to any of these levels, but it cannot have an energy intermediate between two levels. The atom doesn't emit radiation if remains in the same quantum state.

2. An atom can make a transition from one energy level to another level by emitting a quantum of electro-magnetic energy (photon). The energy of an emitted or absorbed photon equals the energy difference between the levels:

,

where [image] J s is Planck's constant, and [image] is the photon's frequency.

3. The third Bohr's postulate assumes that an electron in an atom can revolve in certain circular stable orbits with the angular momentum being quantized:

   (n = 1, 2, 3 ...)

where m is the electron's mass, v is its speed, and [image] is the orbit radius. The integer n is called the quantum number.

In 1924 Bohr's postulates were clearly interpreted in a theory of the French physicist de Broglie. According to him, an electron (or any other microobject) with mass m, moving with speed v, should have a wavelength

.

In the case of an electron's motion in the hydrogen atom, Bohr's postulates result in

.

For the de Broglie wave around a circular orbit to join onto itself smoothly, the circumference of the orbit must be an integral number of wavelengths.

For the hydrogen atom

,

where [image] m is the Bohr radius.

Therefore the third Bohr's postulate is an equivalent to the supposition of existing the standing electron waves on stable circular orbits.

One should emphasize that the notion of an electron moving in atom in circular orbits according to classical laws has just historical meaning. Modern quantum physics describes the electron in atom with the use of so-called wave functions and gives up classical notion about microobject's trajectories.


The Balmer series of Hydrogen

The emission spectrum of an atom can be studied in detail using the instruments called the spectroscope or spectrograph. In simplest spectroscope the light passes through a slit and is then dispersed passing through a prism. A lens system focuses the dispersed light for viewing or onto a photographic plate. Each wavelength of light forms an image of the slit.

The spectrum of a gas is a set of lines of different color. Each line corresponds to particular wavelength of light emitted by the atom.

The origin of line spectra can be understood in general forms on the basis of two central ideas. One is the photon concept; the other is the concept of energy levels of atoms. These two ideas were combined by the Danish physicist Niels Bohr in 1913.

Bohr's theory of hydrogen atom allowed to explain the existing of well-known spectral series called Balmer series in hydrogen spectrum. This series lies in visual range of light and consists of several lines. The line with the longest wavelength, or lowest frequency, in the red, is called [image]; the next line, in the blue-green, is [image]; the next, in the violet, is [image]; and so on.

The Balmer (and other) series suggest that the hydrogen atom has series of energy levels.


Electromagnet

When an electric current flows through a coil of wire, the coil has magnetic field like that of a permanent magnet. The current - carrying coil has a north and a south pole. It can be called an electromagnet. The strength of the field about an electromagnet can be increased many times by placing an iron core inside the coil. Modern technical electromagnets have iron cores of special forms. They are capable to lift very heavy iron bodies.


Electrolysis

Electrolytes are water solutions of inorganic substances (for example, salts in water). Chemical transformations occur in electrolytes when a current passes through them. Such substances are called electrolytic conductors.

The laws of electrolysis established experimentally by Michael Faraday in 1836 played a great part in discovering the discrete nature of electricity. According to these laws, the mass of a substance liberated when a current passes through an electrolyte is proportional to the charge carried by the current


Interference of water waves

The phenomena of diffraction and interference of waves obey the same laws for the waves of different nature. That is why some optical experiments can be modeled by water waves in a ripple tank. For example, one can observe an interference pattern set up by overlapping water waves on the water surface. If the water waves generated by plane mechanical vibrator go trough two narrow slits in a screen. They overlap due to the diffraction and produce the interference pattern like that in optical double-slit experiment of Young.


Lightning

Lightning is a brilliant display of atmospheric electricity. The visible portion of a lightning strike is preceded by an invisible stage in which a column of electrons is extended from a cloud to the ground. These electrons come from the cloud and from air molecules that are ionized within the column. Once the column reaches the ground, electrons within it are rapidly dumped to the ground. During the damping, collisions between the electrons and the air molecules within the column result in a brilliant flash of light. Each strike delivers some 10.000 amperes of electron flow from the cloud base to the ground.


Michelson's Experiment

The speed of light in vacuum is one of the most important physical constants. The first approximate measurement of the speed of light was made in 1676 by the Danish astronomer Olaf Roemer observing eclipses of one of Jupiter's satellites. After that there were many attempts to determine the speed of light more and more precisely.

The most interesting are the direct methods of measurement where the time of a short light pulse passing along a path of given length is measured.

In 1926 american physicist Albert Michelson measured the speed of light at Mount Wilson observatory along a path of 35 km using the rotating mirror method. The main source of error in this method is inconstancy of air refraction index along the path.

In the following years the methods of measuring the speed of light were improved. With the use of laser technology and quantum standards of frequency the precision was managed to be improved in more than four thousand times comparing with Michelson's Experiment.


Nuclear Fission

Nuclear fission is a decay process in which an unstable nucleus splits into two or more fragments of comparable mass. The uranium isotope [image] undergoes fission when bombarded with neutrons. The elements barium and krypton or strontium and xenon are typical results of fission. The total kinetic energy of the fission fragments is enormous, about 200 MeV. Two or three free neutrons usually appear along with the fission fragments. If one or more of these new neutrons cause fission, those fission releases two or three more neutrons, each can cause more fission. This process is called a chain reaction.


Rainbow

Rainbows are created by a combination of refraction and total internal reflection of sun rays in raindrops. As a rainbow is viewed, violet always appears on the bottom and red on the top. Occasionally a secondary rainbow can be seen.

Rainbow is the result of dispersion of white light by water droplets in the air. The light comes from behind the observer, is refracted into a water droplet, undergoes total internal reflection from the back surface of the droplet, and is refracted back to the observer. Dispersion causes different colors to be refracted at different angles. The second slightly larger rainbow has reversed colors. It is created as the result of dispersion and two total internal reflections.