1

Preface

The preparation for the Physics GRE is the undergraduate physics

syllabus. There is no shortcut for it. This manual is not a substitute for the entire

undergraduate physics syllabus. It is instead a collection of important points for

the exam. The reader is advised to learn the material from physics textbooks,

while using this manual for a quick gist of formulae and shortcuts that may be

difficult to locate in a comprehensive textbook. In addition, this manual points out

cases that may appear often in the Physics GRE.

Elaborate derivations are omitted from this manual and final results are

presented. The contents are condensed for easier browsing and quick learning.

Focus is placed on preparing for the Physics GRE rather than comprehensive

mastering of undergraduate physics curriculum.

The manual does not include all materials tested in the Physics GRE.

Readers may solve past questions and refer to textbooks for questions that do not

relate to the contents of this manual.

Despite its limitations, it is hoped that the manual will serve to provide

assistance and guidance to test-takers. It is wished that the reader will be able to

extract positives from it in some capacity.

2

Chapter 0: Basics and Tricks

0.1 Physics GRE Tricks

The Physics GRE tests not only tests knowledge of physics. Solving Physics GRE

problems requires promptness in applying appropriate formulae, shortcuts and

comparisons. The time constraint in the exam compels the examinees to resort to such

tricks, in lieu of elaborate calculations. Also, since answers need to be selected from a

given set of five choices for each problem, it is important to be able to eliminate choices.

Below are some tricks that test-takers are advised to keep at the tops of their heads while

taking the exam:

Recall formulae related to the subject matter of the problem

Remember units of quantities considered; sometimes in problems with seemingly

elaborate calculations, dimensional analysis can be used to eliminate choices

Think of limits – analyse what happens to the quantity under consideration when

certain factors approach zero or infinity

For realistic scenarios, eliminate non-pragmatic answer choices; knowledge of the

magnitudes of factors (such as wavelength of light, speed of electrons in wire,

etc.) can be helpful in estimating scales of answer choices

Mentally convert decimal numbers to nearest fractions for quick mental

calculations; writing on paper consumes time

Imagine graphs of the relevant quantities, without actually drawing them on

paper, to help discern change in one quantity upon change in another

Relate approximations to known quantities, e.g. polygon to circle

3

0.2 Practice for the Physics GRE

Apart from learning the material and solving practice problems, the following exercises

maybe helpful for the physics GRE:

Becoming comfortable with fractions and ratio arithmetic, including quick

additions and subtractions of fractions with different denominators and inverting

fractions

Practising multiplication and divisions of exponents, including roots and powers

of 10

Visualising vector quantities drawn on paper and deducing directions of cross-

products

Identifying important quantities from a written description of the problem for

applying formulae

Memorizing approximations of sin and cos functions at small angles

Understanding shapes of polynomial curves

Getting familiar with units, including base units of quantities like force and

energy, and developing quick dimensional analysis skills; knowledge of base

units of permittivity and permeability may be useful too

4

Chapter 1: Mechanics

1.1 One Dimensional Collisions

Inelastic Collisions

Kinetic energy is not conserved in inelastic collisions. Therefore, in questions involving

inelastic collisions, only momentum is conserved.

Elastic Collisions

Kinetic energy is conserved in elastic collisions. Both principle of conservation of

momentum and principle of conservation of kinetic energy are applicable in such

situations.

Special Case:

A common scenario encountered in the PGRE is the elastic collision of two

bodies, where one is initially at rest and the other is moving at a non-relativistic

speed prior to the collision.

Supposing (in appropriate units):

Mass of moving body = m1

Mass of stationary body = m2

Initial speed of moving body = u

m1 m2

At rest before collision Speed before

collision is u

5

Final speed of m1 = v1

Final speed of m2 = v2

The following formulae (‘shortcuts’) are true:

–

–

The above formulae can be derived by comparing the equations for the

conservations of momentum and kinetic energy for this scenario.

1.2 Rotational Kinetic Energy

Any rolling body possesses both translational and rotational kinetic energies. If the body

does not slip or slide while rolling, then there exists a predetermined relationship

between the rotational and translational kinetic energies, based on the moment of inertia

of the body.

Moment of Inertia

The moment of inertia is the equivalent of mass in rotational motion. Analogous to the

property of mass of resisting linear acceleration, moment of inertia possesses the

property of resisting angular acceleration. The moment of inertia of a body has to be

specified about a definite axis of rotation, unlike mass. The moments of inertia of three

shapes about the centre of mass are mentioned in the Table of Information in the GRE

question booklet:

6

The moments of inertia of a ring or a shell about an axis through the centre, perpendicular

to the plane is:

Rotational Kinetic Energy

Rotational kinetic energy is given by:

For a rolling body, the total kinetic energy is:

7

Problem

A ball is allowed to roll down from rest along the edge of a frictionless, semi-

spherical bowl of radius h. What will be the speed of the ball when it reaches the

bottom of the bowl, if it does not slip or slide?

h

Solution:

The translational speeds of two bodies rolling down from the same height on a

frictionless surface will be equal if the bodies have the same shape, regardless of their

masses and radii.

1.3 Friction on Slopes

When a body is sliding down a rough slope, the frictional force will depend on the weight

(rather, the component of weight normal to the slope). However, the acceleration will be

independent of the mass.

Coefficient of friction

When an object is at limiting equilibrium on a slope at an angle , the coefficient of

friction is given by:

8

1.4 Motion under Gravitational Field

Gravitational field is an inverse-square force. For a planet orbiting the sun in a circle, the

following is true:

Some important points to notice here:

i) The motion is independent of the mass of the planet (true for non-circular

orbits too)

ii) The speed (and hence angular speed) is dependent on mass of sun

iii) (Kepler’s third law)

For elliptical orbits, speed of planet is not constant – it is maximum at perigee and

minimum at apogee.

1.5 Lagrangian and Hamiltonian Equations

For each degree of freedom, the equation of motion for it can be obtained by solving the

Hamiltonian or Lagrangian equation.

9

Chapter 2: Thermal Physics

2.1 Boltzmann Distribution

Molecules at any given temperature can occupy different energy levels. As temperature

increases, higher energy levels are occupied with more and more molecules.

T = 0 K, all molecules occupy the lowest energy level existing.

As T → ∞, all the available energy levels are uniformly occupied (after allowing

for degeneracy).

Partition Function

In a Boltzmann distribution, the fraction of molecules belonging to the i’th energy level is

given by:

where Z is the partition function and gi is the degeneracy of the energy level εi.

When a distribution of molecules is given occupying different energy levels, the partition

function is given by:

2.2 Expansion and Compression of Gas

Work Done BY and ON a Gas

When the internal energy of a gas is decreased through the involvement of work, positive

work is said to be done BY the gas ON the environment, e.g. expansion of a hot gas,

which results in cooling of the gas (i.e. decreases in internal energy) and rotation of a

turbine (i.e. work on the environment).

When the internal energy of a gas is increased through the involvement of work, positive

work is said to be done ON the gas BY the factor which causes the increase in internal

energy, e.g. compression of a cold gas, which results in heating of the gas (i.e. increases

in internal energy), by a bicycle pump (i.e. factor which does the work).

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Work done is given by the formula:

The following pressure-volume diagram illustrates the visual interpretation:

Pressure

W

V1 V2 Volume

For an ideal gas, work done by n moles of gas in an isothermal change is given by:

Some useful shortcuts regarding natural logarithms and work done:

For 0 < x <1, log x is negative (i.e. compression is negative work done by gas)

For x = 1, log x = 0 (i.e. work done is zero if V1 = V2)

For x = 2, log x = 0.69 (i.e. work done is 0.69nRT if volume is doubled)

For x = e, log e = 1 (i.e. work done is 2nRT if volume is expanded 2.71 times)

Work done in an adiabatic change (ΔQ = 0) is always smaller than in an isothermal

change for the same change in temperature.

2.3 Degrees of Freedom

In gases, each degree of freedom contributes

to the . The of the gas is the sum

of the individual contributions of each degree of freedom. Therefore, the of a gas is

the product of the number of degrees of freedom and

.

Area

representing

work done

Upon expanding from V1 to V2:

Work done BY gas =

Work done ON gas =

Vice-versa is true for a compression from V2 to V1

11

Monoatomic Gas

A monoatomic gas is able to move in three directions, and hence possesses three degrees

of freedom.

Diatomic Gas

In addition to the three translational degrees of motion, a diatomic gas at room

temperature is able to rotate on two axes. At higher temperature, the diatomic molecule is

able to rotate on three axes, but the contribution by rotation around the third axis is

negligible.

At even higher temperature, vibration is possible. Vibration adds two degrees of freedom

– one for the translational motion and the other for the potential energy of vibration. This

adds R to .

Consequently, the of a 3D oscillator is 3R.

x

z

y

For a diatomic molecule aligned along the x-axis:

At room temperatures, rotation around the z-axis

and y-axis are permitted.

At higher temperatures, rotation around x-axis is

also possible, but its contribution to Cv is negligible.

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2.4 Entropy

For a reversible process, entropy change is given by:

For an irreversible process,

2.5 Einstein’s 3D Harmonic Oscillator Model

According to this model, atoms in a metal are held together by three spring forces in three

mutually perpendicular directions. The molar heat capacity is given by the formula:

As T increases, C → 3R, which is the universal value of C for metals.

At small values of T, C is given by Debye’s theorem:

13

Chapter 3: Modern Physics

3.1 Relativistic Transformation

Gamma Factor

At relativistic speeds, the following table of shortcuts for the value of gamma will be

useful for quick estimations:

v (m/s) v/c γ

2.997 E8 0.999 22.4

2.985 E8 0.995 10

2.97 E8 0.99 7

2.94 E8 0.98 5

2.91 E8 0.97 4

2.88 E8 0.96 3.6

2.85 E8 0.95 3.2

2.7 E8 0.90 2.3

2.4 E8 0.80

2.1 E8 0.70 1.4

1.8 E8 0.60

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1.5 E8 0.50

1.2 E8 0.40 1.09

0.9 E8 0.30 1.05

Mass, Length and Time Transformations

An object moving at relativistic speed with respect to a stationary reference point

undergoes the following changes in mass, length and time, as observed from the

stationary point:

Problem

If a charged pion that decays in 10-8

s in its own rest frame is to travel 30 m in the

laboratory before decaying, what is the pion’s speed most nearly?

A) 0.43 E8 m/s; B) 2.84 E8 m/s; C) 2.90 E8 m/s; D) 2.98 E8 m/s; E) 3 E8 m/s

Solution:

, i.e. (D)

3.2 Photons

Photons carry quanta of energy. The energy carried by a photon is given by:

15

Photoelectric Effect

In atoms, valence electrons are bound to the atom by a certain amount of energy, called

its work function. When a photon carrying at least that amount of energy strikes the atom,

the electron is ejected.

Compton Effect

When photons are scattered off particles, some of the energy carried by the photons are

absorbed by the particles, while the scattered photons undergo a change in wavelength

given by:

Bragg’s Diffraction

Regular spacing in between crystals can be used to diffract x-rays like a diffraction

grating. For a beam incident on a crystal surface (i.e. Bragg plane) at an angle θ with the

gap between the layers of crystal being d, the following relations are true for interference

of rays reflected off successive crystal layers:

d

Incident ray

Reflected ray from

top layer

Reflected ray from

bottom layer

2d sin θ

θ

θ

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3.3 Mass-Energy Equivalence

The energy content of mass is given by:

For a mass moving at a relativistic speed, the total energy content of the mass is given by:

De Broglie Wavelength

The De Broglie wavelength is the wavelength of a particle when its wave properties are

considered:

For particles with small momentum, wavelength is large, and vice-versa.

Kinetic energy is given by:

Binding Energy of Nuclei

Nuclei are bound by certain amounts of binding energy dependent on the number of

nucleons. The binding energy per nucleon is greatest for iron-56 isotope (~8.8

MeV/nucleon). During nuclear fission and fusion, the binding energy of the new isotopes

formed increase towards the binding energy of iron-56.

Binding

Energy per

nucleon

Mass Number

Fe-56 (8.8 MeV)

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3.4 Atomic Absorption and Emission Spectra

When electrons in atomic orbit absorb photons containing certain energy quanta, they

jump to a higher energy level. Similarly, upon transition of electrons from higher to lower

energy levels, photons are emitted. These photons appear as spectral lines when

observed.

Hydrogen-Like Atoms

In hydrogen-like atoms, i.e. atoms containing a single electron, the energies of photons

emitted by transitions of electrons from infinity (i.e. from very far away from the atom)

to any energy level are related by the following formulae:

Important Point: The energy required to remove the second electron is always greater

than the energy required to remove the first electron. As a result, the total energy required

to remove two electrons is more than double the energy required to remove the first

electron.

Special Case 1: Hydrogen Atom

For hydrogen, Z = 1.

For Physics GRE questions, it might be helpful to memorise the above numbers.

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Also, questions regarding ratios of wavelengths and frequencies of photons can be

answered using the following relations:

Special Case 2 : Positronium

Positronium is similar to a hydrogen atom, except that the nuclear proton is

replaced by a positron.

The following differences between hydrogen atom and positronium are important:

I. The centre of rotation in positronium is located in between the electron

and the positron.

II. The reduced mass of the system is half the mass of an electron.

III. The radius of the circular path in positronium will be half the radius of the

circular path in hydrogen atom. On the contrary, the distance between the

electron and positron will be the same as the distance between electron

and proton in a hydrogen atom.

IV. The speeds of the electron and positron will be the same as the speed of

the electron in hydrogen atom.

V. The energy levels in positronium will be half the magnitude of the

corresponding energy levels in hydrogen atom, i.e. E1 = 6.8 eV.

e- e+

Path traced by motion of

electron and positron

Centre of rotation

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3.5 Wien’s Law and Stefan’s Law

Wien’s Law

Of the radiation emitted from a black body, the wavelength emitted with greatest

intensity is inversely proportional to the temperature of the body.

Stefan’s Law

The power per unit area radiated by a black body is directly proportional to the fourth

power of the temperature of the black body.

20

Chapter 4: Electricity and Magnetism

4.1 Circuits

Resistors

Resistors in Series

When connected in series, the total resistance of the combination is the sum of the

individual resistances.

Resistors in Parallel

When connected in parallel, the reciprocal of the total resistance is the sum of the

reciprocals of the individual resistances.

Important point: The total resistance of a parallel combination is always smaller

than the resistance of the smallest resistor in the combination.

Special Case:

When two resistors of resistance R each are connected in parallel, the total

resistance of the combination is

.

Resistivity

Resistance depends on the resistivity of the material, as given by:

Internal Resistance

Internal resistance is the resistance possessed by the power source, such as a

battery. The energy dissipated at the internal resistance is generally waste heat. If

a resistor of resistance R is connected to the power source, maximum energy is

dropped at the resistor when R equals the internal resistance.

Current

Current through a series circuit is constant at all points of the circuit. When branching is

available, as in parallel circuits, higher current passes through the branch with lower

resistance.

21

Speed of charged particles in a current-carrying wire is related by:

The speed of electrons in a metal wire is in the order of cm/s.

Capacitors

Capacitors in Series and Parallel

When connected in series, the total capacitance of the combination is given by a

similar formula as the one for resistors in parallel:

The total capacitance in a series connection is always smaller than the capacitance

of the smallest capacitor.

For capacitors connected in parallel, the total capacitance is given by the sum of

the individual capacitances.

Charges Stored in Capacitors

The charge stored in a capacitor is given by:

For an a.c. voltage, the amount of charge accumulated in a capacitor depends on

the frequency of the voltage. For high frequency, i.e. small time interval between

voltage switching, charge accumulation in capacitor is small, and vice-versa.

The accumulation of charge opposes flow of current through the capacitor, and

hence reduces the rate of accumulation of charge, creating the exponential

relation:

Accordingly, while a capacitor is discharging, the rate of discharging decreases as

charge content decreases.

Voltage Across Capacitors

The voltage across a capacitor increases as it accumulates charge, and vice versa.

22

Just as with accumulation of charges in a.c. voltage, the voltage accumulation is

small at high frequency and larger at low frequency.

Energy Stored in a Capacitor

The energy stored in a charged capacitor is given by:

Inductors

Inductors oppose the flow of current by virtue of production of magnetic field. Initially,

the voltage across an inductor is equal to the source voltage and flow of current is zero.

As more work is done by the voltage source, current flows through the circuit and voltage

across the inductor drops exponentially.

In a.c. voltage, the voltage drop across the inductor is large for high frequency and small

for low frequency.

RC Circuits

RC circuits consist of a resistor (R) and a capacitor (C). The charge in the capacitor

changes exponentially with time. The quantity RC is the time factor, i.e. after every time

interval equal to the quantity RC, the voltage changes by e (2.71) times.

The half life of this charge decay is RC ln 2.

RL Circuits

The current flow in a resistor-inductor (RL) circuit is exponential and is given by:

23

RLC Circuits

An RLC circuit connected to an a.c. voltage resonates at a certain a.c. frequency. At

resonance frequency, maximum energy is absorbed and stored in the inductor and

capacitor, and current is maximised.

High-pass and Low-pass Filters

A high-pass filter is able to isolate the highest frequency input out of two or more a.c.

voltage inputs of different frequencies. In an RC circuit, the highest frequency voltage

drop would occur at the resistor (voltage drop at the capacitor will be small since high

frequency does not allow large charge accumulation in the capacitor), and hence the

voltage measured in parallel to the resistor will be the high-pass filtered voltage.

In an RL circuit, the voltage drop across the inductor will be the highest frequency (high

frequency will not allow the initial voltage across the inductor, which is also equal to the

highest frequency voltage, to drop much).

The low-pass filter in each of these circuits will be across the remaining component in the

circuit.

4.2 Curl and Divergence

Curl is given by the cross product of the operator, and the divergence is given by the

dot product of the same operator.

Curl of a gradient field is zero. The electric field, which is conservative, is the gradient of

a scalar potential field. Hence,

Divergence of a curl is zero. The magnetic field is the curl of a vector magnetic potential

field. Therefore,

Laplacian

Laplacian is the operator .

24

Divergence Theorem

Curl Theorem

4.3 Electric Fields

Gauss’ Law

The charge enclosed by a Gaussian surface is equal to the surface integral of the electric

field with the area vector of the Gaussian surface times the permittivity.

Using Gauss’ law, the electric field of various uniformly charged shapes can be

deduced:

Important points:

i. The electric field inside a closed REGION containing zero charge is zero.

e.g. field inside a hollow charged sphere, in the absence of any other

charged body, is zero.

ii. The electric field at any POINT is the sum of all the individual electric

field lines present at that point.

e.g. at any point inside a hollow charged sphere, in the absence of any

other charged body, field is zero because the sum of the field lines at that

25

point is zero. However, in the presence of an external charged body, the

field at that point will not be zero, but will be equal to the field strength of

the charged body at that point.

iii. Gauss’ law is true only because electric field is an inverse-square field.

Coulomb’s Law

Coulomb’s law gives the force exerted by a point charge on another point charge.

Curl of Electric Field

Since electric field is a conservative field, in the absence of any changing magnetic field,

the closed line integral of an electric field is zero.

4.3 Electric Potential and Force

Electric potential is the integral of electric field over the distance under consideration.

Consequently, the electric field is the potential gradient.

Potentials of some charged bodies are given below:

The force exerted by an electric field on an external charge is given by:

26

Earnshaw’s Theorem

Earnshaw’s theorem implies that an electrically or magnetically charged particle cannot

be held at equilibrium by the fields of a finite number of other charged particles. The

potential at the location of the particle cannot be a local minimum or maximum, although

it can be a saddle point.

Mirror Charges

The concept of a mirror charge simplifies problems involving a charged body and a

surface at a fixed potential.

e.g. A plane placed at a distance r from a charge q is grounded to maintain a zero

potential. What is the electric field at the midpoint between the charge and the plane?

The situation can be modeled using a mirror charge placed on the other side of the plane

at a distance r, and ignoring the plane altogether. The field at the specified distance is the

sum of the fields of the two charges:

4.4 Magnetic Fields

Right-Hand Grip Rule

For a straight wire carrying a current, the magnetic field is given by the right-hand grip

rule. According to this rule, the wire is gripped with the right-hand with the thumb

pointing in the direction of the current. The direction of the field is given by the direction

of the fingers.

q r r Mirror Charge( –q)

27

Force Between Two Parallel Wires

If two parallel wires carry current in the same direction, then there is attraction between

the wires. If the wires carry current in opposite directions, then there is repulsion.

Biot-Savart Law

The Biot-Savart law gives the magnetic field produced by an electric current.

Curl and Divergence

The divergence of a magnetic field is zero.

This implies that a single point exhibiting properties of a single magnetic pole does not

exist. Magnetic poles must exist in opposite pairs, producing a magnetic field which

starts from the north pole and ends at the south pole. The surface integral of this field

over any closed surface will be zero, since the total number of field lines entering the

surface equals the number of field lines leaving the surface.

The curl of a magnetic field, in the absence of changing electric field, is given by:

This can be used to derive the magnetic fields around different configurations carrying

electric currents:

N

S

Inside the solenoid, the magnetic field is uniform and

points from the south pole to the north pole.

Outside the solenoid, the magnetic field is zero.

28

4.5 Changing Magnetic Field and Flux Linkage

Curl of E and B

In the presence of a changing magnetic field, the curl of electric field is not zero:

The line integral becomes the same as the induced e.m.f. by the changing magnetic field

and consequent flux linkage.

In the presence of changing electric field, the curl of magnetic field becomes:

Induced EMF

Induced e.m.f depends on the rate of change of flux linkage and the number of turns of

coil used to create the flux linkage. And, the induced voltage is formed in a manner

which opposes the cause of the induction (Lenz’s law).

The dot product of the magnetic field with the given area enclosed by the coil during

change in magnetic field is given by:

In certain cases, the area vector might be changing directions. Also, if the angle is a

function of time, θ might be replaced by a function of time, e.g. for rotation at constant

speed, θ=ωt.

In case the e.m.f. is induced in an open-ended configuration, such that no current can be

produced, a p.d. builds up across the system.

29

Special Case:

If a bar moves along at a speed v along conductive rails separated by a distance L

in the presence of perpendicular magnetic field, the induced e.m.f. is given by:

Induced Magnetic Poles and Direction of Current

According to Lenz’s law, the induced e.m.f. is formed in a manner that it resists the cause

of flux linkage. In case of a magnet moving through a loop, the surfaces of the loop

facing opposite directions will be induced into different magnetic poles, such that the

moving magnet experiences an opposing force.

The direction of the induced current can be deduced from the magnetic pole formed, and

vice versa.

When looking from one end of the coil, if the current is flowing in:

I. clockwiSe direction, then the pole is South

II. aNti-clockwise direction, then the pole in North

N S S N

L v

Magnetic Field acting into the

paper or out of the paper,

perpendicular to the direction

of motion of the bar

30

Problem

As in the above figure, a bar magnet travels through a metallic loop connected to

a voltmeter at constant speed. What will be the shape of the graph of voltage

reading against time?

Solution:

The voltages induced as the magnet approaches and leaves the loop will be

constant in magnitude (implied by the constant speed of the magnet) but opposite

in signs, according to the polarity of the closer end of the magnet.

S N

As seen from left end:

aNticlockwise current

curren t

N

S

As seen from right end:

clockwiSe current

S N

V

31

Fleming’s Left-Hand and Right-Hand Rules

In applying Fleming’s left and right hand rules, the following conventions are used

i) The thuMb represents the direction of Motion of loop or wire

ii) The First finger represents the direction of magnetic Field applied

iii) The seCond finger represents the direction of Current, i.e. flow of positive

charge, in the loop or wire

The three fingers, of the left or right hand, are positioned mutually perpendicularly:

Left Hand Rule (Motor Rule)

When the direction of current and direction of applied magnetic field are known

and they are perpendicular to each other, Fleming’s left hand rule can be used to

find the direction of the resultant force, i.e. motion.

The first and second fingers of the left hand should be pointed in the directions of

the field and current respectively, and the direction in which the thumb points will

be the direction in which the current carrying wire experiences force.

V

t

Thumb

First

Finger

Second

Finger

Thumb

First

Finger

Second

Finger

Left Hand: Right Hand:

32

Right Hand Rule (Dynamo Rule)

When the direction of force applied and direction of applied magnetic field are

known and they are perpendicular to each other, Fleming’s right hand rule can be

used to find the direction of the resultant current.

The thumb and first finger of the right hand should be pointed in the directions of

the force and field respectively, and the direction in which the second finger

points will be the direction in which the current will flow.

Problem:

A loop of wire is moving towards a region with an applied magnetic field pointing

into the paper, as in the above figure. What magnetic pole will be induced at the

top surface of the wire loop?

Solution:

Using Fleming’s right hand rule, the direction of the induced current in the bottom

segment of the wire loop is found to be from left to right, i.e. in anticlockwise

direction. Therefore, the induced magnetic pole on the top surface of the loop is

north pole.

Both the left and right hand rules are derived from the formula for the force on a

charged particle by a magnetic field:

X X X X X X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X X X X X X

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4.6 Magnetic Force on Moving Charges

For a charged particle q moving with a velocity in the presence of a magnetic field ,

the force exerted on the charged particle by the magnetic field is given by:

Since current is a flow of charged particles, magnetic force is also exerted on current

carrying wires.

Important Point: The force exerted and the direction of motion is always perpendicular to

the direction of the field. Therefore, the work done by a magnetic field is always zero.

Special Case:

If a current carrying wire is placed in a region with magnetic field perpendicular

to the direction of current, the magnitude of the force exerted on the wire will be:

in a direction given by Fleming’s left hand rule.

Cyclotron

In a cyclotron, a charged particle is made to accelerate in a circular path using magnetic

force as the centripetal force. Since the magnetic field is perpendicular to the path of the

charged particle, the following equation is valid:

The particle is linearly accelerated in the gap between the two hemispheres of the

cyclotron, where the above equation does not apply.

X X X X X X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X X X X X X

L

34

Electric and Magnetic Forces Together

When electric and magnetic forces are simultaneously applied on a charged particle, the

resultant force on the particle is:

When the magnetic and electric forces on the charged particle result to zero, the direction

of the electric field must be given by:

Important Point: If the speed of the particle is varied, only the magnetic force changes. If

charge of the particle is varied, magnitudes of both forces are equally affected.

4.7 Electromagnetic Wave

Light is an electromagnetic wave. In the absence of any changing magnetic field or

changing electric field, the electric and magnetic fields satisfy the two wave equations:

It follows that the speed of light in the medium is given by:

Given the direction of propagation of an electromagnetic vector , the directions of the

electric and magnetic field must be oriented such that:

For a plane-polarised wave, the electric and magnetic fields must be constant in direction

and sinusoidal:

The energy density carried by an electromagnetic wave is given by:

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Chapter 5: Quantum Mechanics

5.1 Wavefunctions

Schrodinger Equation

Locations of particles at quantum energy levels are described using their wavefunctions.

The Schrodinger equation relates the wavefunction of a particle with its energy and

potential applied on it. The time-independent Schrodinger equation, applicable when the

potential is time-independent, is given below:

The solution of the Schrodinger equation gives the shape of the wavefunction under the

given constraints.

Some common wavefunctions, their shapes and energies are listed below:

I. Particle in an Infinite Square Well

II. Harmonic Oscillator Particle

Since the potential outside the square well

is infinity, the wavefunction there is zero.

The potential inside the well is zero, so the

wavefunction is sinusoidal in that region.

n=0

n=1

n=2

n=3

V =

The wavefunctions, belonging to the class

of Hermite polynomials, consist of a

Gaussian function and alternating even

and odd functions.

The energy is given by:

which implies that the energy of the

ground state is not zero.

0 a

n=0

n=1

n=2

n=3

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III. Hydrogen electron

The hydrogen electron wavefunction consists of two functions:

Radial Function (Rnl); describes variation with radius

Spherical Harmonic ( ); describes variation with angles

The first few radial functions are:

The radial probability densities for the above radial wavefunctions are plotted

below:

Important points:

i) The most probable distance of the electron at ground state is

ii) The most probable distance of the electron at ground state is

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Probability of locating a particle in a given region is given by:

For a purely real wavefunction, the complex conjugate of is the same as :

Properties of Wavefunction

A wavefunction must satisfy the following criteria:

i. A wavefunction must be continuous and differentiable everywhere under finite

potential. When travelling from a region of finite potential to a region of infinite

potential, the wavefunction need not be differentiable at the boundaries, though it

still must be continuous, e.g. infinite well boundaries.

ii. The probability of locating the particle over all space (-∞, ∞) must be equal to 1,

i.e. the integral of must be 1. In spherical coordinates, the integral of

must be 1 after inserting an appropriate normalisation constant.

iii. The wavefunction must approach zero as distance increases:

Energy of Wavefunctions

The energy of a wavefunction is the kinetic energy of the particle it represents. A

sinusoidal wavefunction carries positive kinetic energy and an exponential wavefunction

carries negative kinetic energy.

i. The curvature of a sinusoidal wavefunction increases with the kinetic energy.

Higher speed of particle implies higher kinetic energy and greater curvature of its

wavefunction.

ii. Higher speed also indicates lower probability of locating the particle in a given

finite region. This is indicated by the greater curvature of the wavefunction, which

gives a lower value of the integral of .

38

iii. When a particle enters into a region of higher potential, its curvature will

decrease. If the potential is greater than the kinetic energy of the particle, the

wavefunction will become exponentially decaying.

5.2 Inner Product and Expectation Value

The inner product of two functions is the integral of the functions:

The expectation value of an operator is the mean value of the operator:

Important point: Different wavefunctions of the same particle corresponding to different

energy levels are all orthogonal, e.g. of particle in a box, harmonic

oscillator, hydrogen electron, etc. are all orthogonal. Therefore, their inner products are

zero.

e.g. if ,

then

E

V

V

E

E > V E < V

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Adjoint and Hermitian Operators

If A is an operator, then its adjoint operator (written as A†) satisfies the following

relationship:

Some properties of adjoint operators are listed below:

Matrix Notation

Operators can be represented using matrices. In such a case, the inner product

becomes the matrix product.

To find the adjoint of a matrix operator, the matrix must be transposed and the

complex conjugate of each entry must be taken.

e.g. The adjoint of

is

When an adjoint is equal to the operator, then the operator is called Hermitian (or self-

adjoint).

Facts about Hermitian operators:

i. Hermitian operators relate to observable quantities, e.g. position, momentum,

energy.

ii. The eigenvalues of Hermitian operators are always real.

iii. A real Hermitian matrix must be equal to its transpose, i.e. be symmetric. A

complex Hermitian matrix need not be symmetric; its transpose has to be equal to

its complex conjugate.

5.3 Commutators

For two operators A and B, the commutator is defined as:

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When two operators are said to commute with each other, their commutator is zero.

Properties of commutators:

5.4 Quantum Numbers

Pauli’s Exclusion Principle

In an atom, the position of every electron is defined by its unique set to quantum numbers

for that atom. Pauli’s exclusion principle states that no two electrons can have the exact

same four quantum numbers.

Quantum Numbers

The four quantum numbers are:

Principal Quantum Number (n)

It indicates the shell number of the electron, starting from n=1 and increasing.

Angular Momentum Quantum Number (l)

It represents the sub-shell number of the electron. For a shell numbered n, there

are (n+1) sub-shells. It starts from l=0 and goes up to l=n-1. The energy levels at

different sub-shells of the same shell are usually different, although in case of

hydrogen atom they are degenerate.

Magnetic Quantum Number (ml)

It represents the orbital number of the electron. For a sub-shell numbered l, there

are (2l+1) orbitals. It starts from ml=-l and goes up to ml=l, inclusive of ml=0. In

the absence of any strong magnetic field, the orbitals are all degenerate.

Spin Quantum Number (ms)

It represents the spin (intrinsic angular momentum) of the electron. In an orbital,

which can accommodate two electrons, there cannot be more than one electron

with the same spin. It can only have values of

and

, also called

spin-up and spin-down.

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5.5 Spin

Spin of electrons was discovered through the Stern-Gerlach experiment. When an atom

with a single valence electron is subjected to an inhomogeneous magnetic field, two

distinct fringes are observed, corresponding to the spin-up and spin-down states of the

valence electron.

In matrix notation:

The spin components in the x, y and z directions are represented by the Pauli spin

matrices:

Fermions and Bosons

Particles with half-integer spins (1/2, 3/2, etc.) are called fermions, e.g. electrons,

protons, neutrons.

Particles with integer spins (0, 1, 2, etc.) are called bosons, e.g. photons, pions.

Adding Angular Momenta

Electrons in an atom have both orbital and spin angular momenta. The total angular

momentum is the vector sum of the two momenta:

The value of j ranges from |l-s| to (l+s), changing by 1 for each state.

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5.6 Stark and Zeeman Effects

Applying an electric field on an atom is analogous to applying a nuclear charge

perturbation. On a hydrogen atom, owing to the electric field, the atom aligns itself to the

direction of the field, i.e. becomes polarized. The change in energy is proportional to the

square of the electric field:

This effect is called the quadratic Stark effect.

For an excited hydrogen atom, the change in energy is proportional to the electric field

(not on its square), and the effect is known as linear Stark effect.

When a magnetic field is applied on a hydrogen atom, the magnetic moments of the

orbital angular momentum and spin angular momentum are affected. This is called the

Zeeman effect. The change in energy is given by:

For the strong-field Zeeman effect, the ml degenerate states separate into different energy

levels (e.g. l = -1, 0, 1) and ms states are added to each different ml state. The resultant

states are the new energy levels.

For weak fields, the different mj states separate into different energy levels.

Selection Rules

Transitions in an atom follow the following selection rules:

5.7 Two Electron Systems

Since electrons are fermions, the wavefunctions of electrons must be anti-symmetric

under particle interchange. Consequently, electrons follow the Pauli’s exclusion

principle, and hence all electrons in an atom cannot drop to the ground state.

43

In an anti-symmetric wavefunction, the sign of the wavefunction must be changed when

the two particles making up the wavefunction are interchanged, e.g.

By interchanging m and n, we get .

Symmetric wavefunctions, such as the wavefunction of a collection of bosons, are

unchanged when particles are interchanged, e.g.

and

which are both unchanged upon interchanging m and n.

Spin Wavefunction

The total wavefunction is made up of the spatial wavefunction and the spin wavefunction.

When both of them are symmetric or anti-symmetric, the total wavefunction is

symmetric. When any one of them is symmetric and the other anti-symmetric, the total

wavefunction is anti-symmetric.

Singlet and Triplet Electron Spin States

A singlet state consists of a single ms value, ms = 0. This occurs when s = 0. In a two-

electron system, it means that the spins of the two electrons must be opposite, such that:

This implies that a singlet state has anti-symmetric spin wavefunction, and hence

symmetric spatial wavefunction.

In a triplet state, ms = -1, 0, 1. This occurs for s = 1, and is brought about by two

electrons with same spins:

The triplet state has symmetric spin wavefunction and hence anti-symmetric spatial

wavefunction.

44

Chapter 6: Optics

6.1 Concave Lenses

A concave lens bends all light that enters towards its focus.

Only the rays parallel to the horizontal axis of the lens reach the focus. The rays not

parallel to the axis are diverted closer or farther away from the focus, depending on the

angle of incidence on the lens.

Focus

Focus

Horizontal Axis

Focus

Horizontal Axis

Focus

Horizontal Axis

45

Some properties about images formed by concave lenses:

For parallel rays, i.e. rays coming from an object at infinity, the image is real and

formed at the focus.

For an object located at a distance greater than 2f (f is the focal length), the image

is real and formed between f and 2f on the other side.

For an object located at 2f, the image is real and formed at 2f.

For an object located between f and 2f, the image is real and formed at a distance

greater 2f.

For an object located at f, the image is formed at infinity.

For an object located at a distance below f, the image is virtual and observed at f.

The following table summarises the points:

Note: For a single lens arrangement, any real image formed must be located beyond f. For

more than one lens, the image formed may be located at a distance smaller than f.

All real images formed are inverted but the virtual image observed is not inverted.

Magnification

For a single lens, magnification is given by the ratio of the image height to object height,

or by the ratio of the image distance to object distance.

Important Point: If a lens is broken into a smaller piece, the properties of the image

formed remain unchanged except for the brightness. The brightness of the image

decreases since fewer rays of light can enter the lens.

Horizontal

Axis

∞

f 2f

At f

Between f

and 2f Beyond 2f

At 2f At ∞

Virtual Image

observed at f

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Resolution

In astronomical telescopes, the resolution of the lens allows the viewer to distinguish

between objects with a minimum angular difference between them. This angle is related

to the diameter of the lens and wavelength of light using the formula:

6.2 Pin-Hole Camera

A pin-hole camera uses light entering through a pin-hole to produce an image inside a

dark enclosure. For a certain diameter of the pin-hole, the image formed has the

minimum blur, owing to minimised diffraction at the smallest possible pin-hole diameter.

This diameter is given by:

6.3 Single Slit and Double Slit Interference

When coherent light is projected through a single aperture, the resultant pattern observed

on a screen is described by the following formulae:

When using double slits, the pattern is described by:

For multiple slits, such as a diffraction grating, the equation is:

For light incident at an angle , on a diffraction grating, it follows:

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6.4 Reflection at Thin Surfaces

When light is reflected and refracted at a surface, the following are true:

If the refractive index of the surface is greater than the refractive index of the

medium through which the incident light propagated, then the phase of the

reflected light will be inverted. If not, then the phase of the reflected light remains

unchanged.

The phase of the refracted light remains unaltered, regardless of the refractive

indices.

The speed and wavelength of the refracted wave changes according to refractive

index, whereas the frequency remains constant. For transition from region of low

to high refractive index, both speed and wavelength decrease, and vice-versa.

When light is incident on a thin surface and reflected off both the top and bottom

surfaces, the reflected waves off both surfaces may interfere constructively or

destructively, depending on the relative refractive indices, surface width and wavelength

of the incident light.

Case 1: Dense surface, with lighter material on both sides (e.g. soap film)

Dense material

Air

Air

The waves are reflected off both top and bottom

surfaces, but phase is inverted only at the top

surface, not at the bottom.

Constructive interference:

Destructive interference:

where is the wavelength inside the material,

given by:

d

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Case 2: Dense surface, with denser material below (e.g. oil on glass)

6.5 Polarised Light

A beam of light may contain waves oriented in different directions, all of which are

perpendicular to the direction of propagation. By polarising, waves of a particular

orientation only can be filtered out. Polarisation can be carried out using a polariser,

which produces plane-polarised light.

When plane-polarised light is filtered through a polariser at an angle θ, the following

changes occur:

The orientation of the filtered light is rotated by an angle θ.

intensity of the filtered light is given by:

For plane-polarised light passing through two polarisers placed at 450 to each other and

the second lens being perpendicular to the direction of polarisation of the light, the

intensity of the light emerging at the end is a quarter of the original intensity.

Dense material

Air

d

Denser material

The waves are reflected off both top and bottom

surfaces, and phase is inverted upon reflection at

each surface.

Constructive interference:

Destructive interference:

where is the wavelength inside the material,

given by: