Ch 1 physics at a glance tủ tài liệu training

PHYSICS AT A GLANCE
DIRECT THEORY NOTES
1. UNITS AND DIMENSIONS
Physics is the science of natural phenomena. It is the science of observation and measurement. It is the
science of interpretation of results. To measure a quantity we need units. To define and establish units, we
require physical standards.
The important requisites of a physical standard are accessibility, accuracy and invariance. Such requisites
are possessed by modern standards. An example of physical standard is the wavelength of light, which is
used to define metre.
In the international system used today in science and engineering, we have seven base units and three
supplementary units. The system is called SI. Table 1.1 gives ten fundamental quantities, i.e. seven base
units and three supplementary units, their symbols and what they represent.
TABLE 1.1 FUNDAMENTAL QUANTITIES
Quantity

symbol

What it represents

metre

m

Length

kilogram

kg

Mass

second

s

Time

kelvin

K

Temperature

candela

cd

Luminous intensity

ampere

A

Electric current

mole

mol

Amount of substance

radian

rad

Angle in plane

steradian

sr

Solid angle

curie

Ci

Radioactivity

Conventions Followed in Using SI units
1. Symbols should not be used with plural, e.g. 10 m is correct, but 10 ms in wrong.
2. A void using bars in writing units, e.g. write m/s2 as ms-2
3. As far as possible multiples of 103n of base units are to be used. Each multiple has a prefix. E.g. when n
= 1, the prefix is kilo. Symbol is ‘k’. When n = 2, prefix is mega; symbol M. When n = -1, prefix is ‘milli’
and symbol m. n = -2, micro etc.
Dimensions:
Physical quantities are classified into fundamental and derived quantities. The five fundamental quantities
in physics are mass (M), length (L), time (T), temperature (θ) and current (I). All others are derived
quantities.
Dimensions are the relation between fundamental and derived quantities. If we write any derived quantity
as Ma Lb Tc θd Ie, then this is called a dimensional formula and the powers a,b,c,d,e are called dimensions.

The dimension of the quantities will be given along with their definitions.

2. MOTION IN ONE AND TWO DIMENSIONS
Scalars and Vectors
Physical quantities are classified into scalars and vectors.
A scalar is fully defined and understood, when its magnitude or value is given, e.g. mass, length, distance,
speed, etc .
A vector quantity is fully defined only when both its magnitude and direction are given.e.g., force,
acceleration, magnetic field etc. All quantities having magnitude and direction will not be vectors. They
must also obey the rules of geometric addition.
Vector Algebra
Two vectors can be added by (i) triangle method, (ii) parallelogram method, and (iii) analytical method.
The sum of two vectors A and B at an angle α is the vector C, where the magnitude of C is given by the
analytical method as
C=

A 2 + B 2 + 2AB cos α

and the direction of C with vector A, θ is given by


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B sin α
A + B cos α
To subtract a vector we reverse the vector to be subtracted and add with the other, e.g. A-B = A+(-B).
To add a number of vectors we use the law of polygon of vectors. If we represent the vectors as the sides
of a polygon, the side that completes the polygon in the opposite direction is the sum.

Multiplication of vectors: Two vectors A and B can be multiplied in two ways. If the product is scalar, it
is scalar multiplication or dot product.
The dot product of two vectors a and B is defined as A.B = A B Cos α where α is the angle between them.
If the product is a vector, it is called a vector product or cross product. The cross product of A and B is A x
B = AB sin α nˆ is unit vector perpendicular to the plane of A and B.
Resolution of a vector: A vector F inclined at angle α with a given direction can be resolved as two
rectangular components, as F cos α along the direction and F sin α perpendicular to it.
Unit vectors: A vector F can be written F = ˆiF + ˆj F + kˆF where ˆi , ˆj and kˆ are known as unit vectors

tan θ =

x

y

z

along the three axes of coordinates. The magnitude of each is one unit. Fx, Fy, Fz are the x,y and z
components of the vector F.
Types of Motion

One-dimensional motion means a particle moves such that its position can be represented by one
coordinate. Two- dimensional motion (motion in a plane) means the position of a particle can be
represented by two coordinates (motion of a projectile). In three-dimensional motion, the position of a
particle can be represented by three coordinates. (motion of molecules of a gas)
Displacement is the change in the position of a body in a given direction. It is a vector.
The rate of displacement is velocity. The rate of travelling a distance is speed
Unit : ms-1 . Dimensions : M0L1T-1
Acceleration is the rate of change of velocity
Unit: ms-2. Dimensions: M0L1T-2

Equations of motion are equations connecting various parameters of a body’s motion. For simplicity we
use conventional letters. If ‘u’ is the initial velocity, ‘v’ the final velocity, ‘a’ the uniform acceleration, ‘s’
the distance travelled, and ‘t’ the duration of travel and ‘sn’ the distance travelled in n th second, then
v = u + at
⎡u + v⎤
s=⎢
⎥t
⎣ 2 ⎦
1
s = ut + at 2
2

v2 = u2 + 2as

1⎤
⎡
sn = u + a ⎢n − ⎥
2⎦
⎣

Equations of motion under gravity (bodies moving vertically up and down under gracvity) are got by
substituting a = + or – g depending on whether the body is travelling downwards or upwards. g is usually
taken as 9.8 ms-2. For rough calculations it can be taken as 10.
Projectile motion: A projectile is a body thrown at an angle so that it moves in a vertical plane under the
action of gravity. If h is the maximum height, R is the horizontal range, T is the time of flight, Rm is the
maximum range, u is the initial velocity of projection and α is the angle of projection of the body
u 2 sin 2 α
2g
2u sin α
T=

g
h=

R=

u 2 sin 2α
g

u2
g
Uniform circular motion: A body in uniform circular motion has constant speed and varying velocity.
Rm =

The acceleration towards the centre of the body is

v2
or ω2 r, where v is its velocity and ω angular
r

velocity. The force towards centre acting on the body is the centripetal force =

mv2
or mω2r
r


Physics at a Glance

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Centrifugal force: An observer in a circular motion feels a radially outward force. This force is called the
centrifugal force. Its value is equal to the centripetal force. It is pseudo a force. The properties of pseudo
force are given in indirect theory notes. .
In a non-uniform circular motion, speed and velocity change. An example such a motion is a stone tied to
the end of a string and whirled in a vertical circle. At the highest point, when the string has minimum
tension (T1)
mv1 2
− mg
r
where v1 is velocity of the stone at the highest point. If the tension is zero, velocity of the stone is

T1 =

v1 =

rg

At the lowest point it has maximum tension T2 given by
mv 2 2
T2 = mg +
r
where v2 is velocity of stone at the lowest point. The velocity of the stone at the lowest point when the
tension at the highest point is zero, is
v2 =

5rg

The difference in tension at the highest and lowest points is
T2 – T1 = 6mg
i.e., 6 times the weight of the stone

3. LAWS OF MOTION
Frame of Reference:

A frame of reference is a coordinate system. It has an origin and axes of coordinates. Two frames, which
move with respect to each other in uniform speed in a straight line, are called inertial frames. Newton’s
first law (law of inertia) is obeyed in such frames.
Two frames having relative acceleration are non-inertial frames. Newton’s second law is obeyed in such
frames
Newton’s Laws of Motion:
First law: A body at rest or moving with uniform speed in a straight line continues so until an external
force acts on it.
Second law: The rate of change of momentum is directly proportional to the unbalanced (resultant) force
acting on the body
Third law: For every action there is an equal and opposite reaction
Newton’s first law defines force while the second law measures it as f = ma, where m is mass and a the
acceleration.
Unit of force: newton (N). Dimensions of force: MLT-2
Weight of a body is a force exerted by the earth on the body: W = mg. It is measured by the reaction,
resistance or tension.
In a freely falling body, only the force of gravity acts. That is, the reaction or resistance is zero. Hence a
freely falling body experiences weightlessness.
Law of Conservation of Momentum:
A closed system is one, in which no external force acts .The total momentum of a closed system remains
constant
Examples: Rocket propulsion, recoil of a gun, explosion of a shell and collision.
For recoil of a gun, mv = MV, where m is the mass of the shot, M is the mass of the gun, v is the velocity
of the shot and V is the recoil velocity of the gun. More accurate formula will be
mv = (M-m)V. For m << M, it can be written as
mv = MV.

For collision in a straight line, by the law of conservation of momentum, we have
m1 u1 + m2 u2 = m1 v1 + m2 v2
where m1, m2 are the two colliding masses, u1, u2, their initial velocities and v1 v2 their final velocities.
Coefficient of restitution: The ratio of the relative velocity of two bodies after impact to that before
impact is defined as the coefficient of restitution e.
v − v2
e= 1
u1 − u 2
For collision between a fixed plane and a body,
e=

v
i.e., ratio of velocity after impact to that before impact. In terms of height,
u


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e=

h2
h1

where h1 is the height of rebound and h2 is the height of fall.
If e = 1, the collision is perfectly elastic. If e = 0, the collision is perfectly inelastic. If e is between 0 and 1
the collision is inelastic. Ordinary mechanical collisions are inelastic. In an elastic collision, both
momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic
energy is lost.

The kinetic energy lost in an inelastic collision is
∆E =

1
2

[

⎡ m1m 2 ⎤
2
⎥ 1− e
⎢
m
m
+
2⎦
⎣ 1

][ u − u ]
1

2

2

where u1, u2 are their initial velocities of masses m1, m2 respectively and ‘e’ coefficient of restitution.
Friction: This is a force, which always opposes relative motion between two surfaces. The value of
frictional force is µa mg, where µs is called coefficient of static friction. When the body is in motion, the
frictional force acting on it is the dynamic frictional force µk mg. The coefficient of static friction is the
ratio of force of static friction to normal reaction. i.e., Fs /R . The coefficient of dynamic friction does not

depend on the speed of motion as long as the speed is constant. The angle of friction λ is the angle between
the resultant reaction and normal reaction. It is related to µs by the equation
tan λ = µs
A body placed on a rough inclined plane is in equilibrium until the angle of plane θ is equal to the angle of
friction λ. Thus the equilibrium condition on a rough inclined plane is
µs = tanθ = tan λ

4. WORK, POWER AND ENERGY
The work done by a force = f s cosθ, where f is the force, s the displacement, θ the angle between the force
and the displacement. W = f. s
Unit of work : joule (J). Dimensions of work : ML2T-2
The rate of doing work is called power. P =

work dW
.
=
time
dt

Unit: watt (W). Dimensions : ML2T-3
A body has energy if it can do work. Energy is measured by the work a body can do. Kinetic energy is the
energy due to motion and is given by 1/2 mv2 where m is the mass and v is the speed. Potential energy is
the energy due to position or state of strain. Potential energy due to position is given by mgh and potential
energy due to state of strain of a spring = 1/2 kx2 where k is the force constant and x is the extension of the
spring.
Work done = energy
fxs=

1
mv2 or f x s = mgh

2

Power = force x velocity, i.e., P = f.v

5 CENTRE OF MASS AND ROTATIONAL MOTION
Centre of mass is a point where the whole mass of the body is supposed to be concentrated. To find centre
of mass of a system of two particles of masses m1, m2 at distances x1 and x2 from the origin (which we can
choose conveniently)
m x + m2x 2
xcm = 1 1
m1 + m 2
The centre of mass in terms of vector distance r is given by
rcm =

m1 r1 + m 2 r2
, where r1 and r2 are position vectors of the two masses.
m1 + m 2

Velocity of centre of mass is given by the equation
m v + m2v2
vcm = 1 1
, where v1 and v2 are velocity vectors of the two masses.
m1 + m 2
Acceleration of centre of mass is given by the equation
acm =

m1a 1 + m 2 a 2
, where a1 and a2 are acceleration vectors of two masses.
m1 + m 2

In the absence of external force, the centre of mass is at rest or moves with the uniform speed. The above
equations can be extended to find the centre of mass of a system of more than two particles
The angular momentum of a particle about the origin is


Physics at a Glance

5

L=rxp
where r is the radius vector and p is the momentum vector
The torque acting on a particle about the origin is
G G G
τ = r xF

where r is the radius vector and F is the force vector
The magnitude of the torque = rF sin θ where θ is an angle between the radius vector and the force vector.
The moment of inertia of a particle about an axis is the product of mass and square of distance from the
axis (mr2). The moment of inertia of a rigid body is I = MK2 where K is the radius of gyration.
The theorem of parallel axis states that
Iz = Icm + Ma2
where Icm is the moment of inertia about an axis through centre of mass perpendicular to its plane, Iz is the
moment of inertia about a parallel axis at a distance ‘a’.
The theorem of perpendicular axis states that
I x + I y = I z,
where Ix, Iy are moments of inertia about two mutually perpendicular axes lying in the plane of a lamina, Iz
is the moment of inertia about an axis perpendicular to its plane.
The angular momentum of a rigid body
L = Iω , where ω is the angular velocity.
The torque acting on a rigid body

τ = Iα, where α is the angular acceleration.
The law of conservation of angular momentum states that the total angular momentum of a body remains
constant in the absence of external torque.
The kinetic energy of rotation of a rigid body is given by
Erot =

1 2
Iω , where I is moment of inertia and ω is angular velocity.
2

For a body having translatory and rotatory motion, the total kinetic energy is given by
E(total) =

1
1
mv2 +
Iω2.
2
2

The angular kinetic energy or the work done by the torque is given by
1
I ω2 = τxθ,
2

This equation is equivalent to

1
mv2 = f x s, in translatory motion.
2

The acceleration of a body rolling down an inclined plane is

g sin θ
⎡k2 ⎤
1+ ⎢ 2 ⎥
⎢⎣ r ⎥⎦

where k is radius of gyration and θ is the angle of the plane.
Kepler’s Laws of planetary Motion:

First law : Each planet moves around the sun in an elliptical orbit, with the sun at one of the foci
Second law: The areal velocity of the planet is a constant, i.e. the radius vector joining the sun to the planet
sweeps out equal areas in equal intervals of time.
Third law: The square of the period of the planet around the sun is proportional to the cube of the mean
distance from sun. i.e. T2 ∝ R3

6. GRAVITATION AND SATELLITES
Newton’s law of gravitation states that the gravitational force between two particles of masses m1 and m2 at
a distance r acts along the line joining them, and has a magnitude
G m1m 2
F=
r2
where G is the gravitation constant
Unit of G: Nm2 kg-2. Dimensions of G: M-1L3T-2
The gravitational field at a point due to a mass m is the ratio of force to mass and is given by
Gm
E= 2
r
Unit: Nkg-1. Dimensions: LT-2 (that of acceleration)

The gravitational potential at a point due to mass m at distance r is given by
V=

−G m
r


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Unit: Jkg-1. Dimensions: L2T-2
The gravitational potential energy between two masses m1, m2 at a distance r is
−G m1m 2
U=
r
The relation between acceleration due to gravity g and gravitation constant G is
G M
g=
R2
where M is mass of the earth and R is the radius of the earth
The relation between field and potential is that E = -

dV
V
or simply
dr
r

The mean density of earth is given by

3g
ρ=
4πRG
Variation of g: The value of g at a height h from the ground is
gR 2
GM
g1 =
=
(R + h ) 2
(R + h ) 2
The value of g at a depth x from the surface of earth is
g(R − x )
g2 =
R
The value of ‘g’ varies with latitude. It has the maximum value at the poles where latitude is 900 and
minimum value at the equator where latitude is 00.
The value of ‘g’ varies with rotation of the earth. It decreases if the earth rotates faster and increases if the
earth rotates slower.
If the earth rotates with 17 times the present angular velocity, g at equator will be zero. The period of the
earth when g is zero at the equator is nearly 84 minutes and is given by
R
T = 2π
g
where R is the radius of the earth
The velocity of an earth satellite going very close to the earth is given by
GM
= gR
v=
R
This velocity is known as the first cosmic velocity.

The velocity of an earth satellite orbiting at a height h from the surface of the earth is given by
GM
v=
R+h
The period of a satellite at a distance r from the centre of the earth is given by
4π 2 r 3
GM
For a satellite going very close to the earth, the time period is given by

T=

T=

R
4π 2 R 3
= 2π
g
GM

The velocity of a body to escape from the earth’s gravitational field is given by
2GM
v=
= 2gR
R
This is known as the second cosmic velocity. It does not depend on the direction of projection.
The period of geo-stationary satellites is one day (24 hours). The distance of such a satellite from the
centre of earth is nearly 42,000 km or from the surface of earth 36,000 km

7 WAVE MOTION AND OSCILLATIONS
A wave is a disturbance set up in a given medium from equilibrium condition.

A progressive wave is one which progresses or travels in a given direction unobstructed. It is a travelling
wave, which carries energy.
Different forms of an equation of a progressive wave:


Physics at a Glance
x⎞
⎛
y = a sin 2πf ⎜ t − ⎟
v⎠
⎝
⎛ t x⎞
y = a sin 2π⎜ - ⎟
⎝T λ⎠
2π
y = a sin
( vt − x )
λ

7

y = a sin (kx - ω t)
where y is the transverse displacement, a amplitude, f frequency, T period, λ wavelength and x is position
of the particle at the time t.
A stationary wave is superposition of two progressive waves in opposite directions. Equation of a
stationary wave:
2π x ⎞
⎛
y = ⎜ 2a cos
⎟ sin ω t

λ ⎠
⎝
Wave frequency (f): Wave frequency is the number of vibrations per second made by the source or particle
in the medium.
Period (T) The period T of wave motion is the time for one complete oscillation.
Wavelength (λ): Wavelength is the distance between two particles in the medium which are in the same
state of vibration.
Wave speed (v): Wave speed is the distance travelled by the wave in one second.
Wave number (v): Wave number is the number of waves in unit length. It is reciprocal of wavelength.
Wave vector (k): Wave vector or propagation constant is 2π/ λ or 2 π v
When two waves of the same frequency and amplitudes a1 and a2 and phase difference φ superpose the
resultant amplitude A is given by the equation
A=

a 1 2 + a 2 2 + 2a 1 a 2 cos φ

The physical effect of superposition is interference
Simple Harmonic Motion

A body is in S.H.M if a restoring force proportional to displacement acts on it.
The equation of simple harmonic motion is
f = -kx
where k is a constant. k is the restoring force for unit displacement.
Acceleration is proportional to displacement of S.H.M
a = -ω2 x where ω is the angular frequency of motion.
Displacement of S.H.M is given by
x = A sin (ω t + φ)
where φ is initial phase.
Period of a spring = 2π

m
, where, m is the mass of suspended body and k is the force constant of the
k

spring.
Period when two springs are in series = 2π

m(k 1 + k 2
, where k1, k2 are force constants of the springs
k 1k 2

Period when two springs are in parallel = 2π
Period of a simple pendulum = 2π

m
k1 + k 2

1
for small angular amplitudes. Here l is length of the pendulum
g

Period of a test tube floating in a liquid = 2π

m
, where, m is mass of the test-tube, A corss sectional
Adg

area, d density of liquid.
Period of a liquid oscillating in a U-tube = 2π

L
, where, L is total length of the liquid in the U-tube.
2g

Period of a body dropped in a tunnel diametrically dug across earth = 2π
earth.

R
, where, R is radius of the
g


Physics for IIT-JEE Screening Test

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Period of the longest pendulum suspended in the vicinity of earth = 2π

R
g

Table 1.2 gives all properties of S.H.M
Here a is the amplitude of S.H.M, x is the displacement, ω is the angular frequency i.e. ω = 2πf, f is the
frequency of S.H.M., T is the period of S.H.M., v is velocity of S.H.M, and a is the acceleration of S.H.M
TABLE 1.2. Properties of Simple Harmonic Motion

Position of the particle
Quantity

At equilibrium

position

At extreme
position

Intermediate
position

0

A

Between 0 and A

ωA

0

Acceleration a

0

-ω2A

-ω2x

Kinetic Energy

1
mω2A2

2

0

1
mω2(A2-x2)
2

0

1
mω2A2
2

1
mω2x2
2

1
mω2A2
2

1
mω2A2
2

1
mω2A2
2

Displacement
Velocity v

Potential Energy
Total Energy

ω A2

x2

Transverse Vibration of Strings

Velocity of transverse waves in a stretched string is given by
T
µ

v=

where T is stretching tension, µ is linear density, i.e., mass per unit length.
The laws of transverse vibration are relations amongst various parameters of a vibrating string. If f is
frequency, µ linear density, l vibrating length, T stretching tension, the laws are
1
f α when µ and T are cons tan ts
l
f α T when µ and l are cons tan ts
1
fα
when l and T are cons tan ts
µ
The frequency of transverse vibration is given by the equation

f=

n
2l

T
µ

for nth mode, i.e., when the string vibrates with n loops. If r is the radius of the wire and d is density of the
wire, the frequency can also be written as
n
T
f=
2l πr 2 d
The fundamental mode corresponds to n=1.
Vibration of Air Column and pipes

A closed pipe is one which is closed at one end. An open pipe is one which is open at both ends. The
fundamental frequency of a closed pipe is given by
V
.
f=
4L
The fundamental frequency of open pipe is given by
f=

V
2L

Physics at a Glance

9

where L is length of the pipe. The closed pipes will produce only odd harmonics f, 3f, 5f…etc., while the
open pipe will produce all harmonics i.e. odd and even f, 2f, 3f......,etc.
Velocity of Sound in Air

Sound waves are basically longitudinal. Velocity of sound waves in a medium depends on elasticity and
inertia of the medium. Velocity of sound in a solid is given by
v=

E
ρ

where E is Young’s modulus and ρ density of the solid.
Velocity of sound in liquid is given by
v=

K
ρ

where, K is Bulk modulus and ρ density of the solid
Velocity of sound in a gas is given by
v=

γP
ρ

γRT
M

=

where γ is ratio of specific heats , T temperature and M molecular mass.
Velocity of sound does not depend on pressure. It is directly proportional to square root of Kelvin
temperature. Velocity of sound increases with humidity. Velocity of sound also changes with motion of
the medium, i.e. wind.
Doppler Effect

The apparent change in frequency of light or sound wave due to relative motion of source or the observer
is called the Doppler effect.
Equation for apparent frequency when the listener and source approach each other with velocity UL and Us
respectively is given by
⎡ V + UL ⎤
⎥
⎣ V − Us ⎦

f ′′ = f ⎢

where V is velocity of sound in air.
In this equation, if listener is at rest, we put UL = 0. If the source is at rest, we put US = 0. If the listener is
receding, we put UL = -Us.
The apparent frequency of light wave of speed ‘c’ from a source of speed ‘v’ given by
⎡

v⎤

⎣

⎦

f ′ = f ⎢1 + ⎥ when the source approaches. When the source recedes
c
⎡ v⎤
f ′ = f ⎢1 − ⎥ .
⎣ c⎦

Generally Doppler effect equation in light can be written as
∆f ∆λ v
=
=
f
λ
c
where ∆λ is the change in wavelength and ∆f change in frequency.
The increase in frequency (or decrease in wavelength) is violet shift. The decrease in frequency (or
increase in wavelength) is red shift.

8. PROPERTIES OF MATTER
Stress in force per unit area. Unit: Nm-2. Dimensions : ML-1T-2
Strain is the ratio of change in dimension to original dimension. It has no unit, no dimensions.
Hook’s law states that stress is proportional to strain
Stress
= Constant = Modulus of elasticity
Strain

There are three moduli of elasticity. Young’s modulus E (or Y) is given by
Longitudinal stress

E=
Longitudinal strain
Bulk modulus K ( or B) is given by
Volumetric stress
K =
Volumetric strain
Shear modulus n (or G) is given by
Shearing stress
n=
Angle of shear


Physics for IIT-JEE Screening Test

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Poisson’s ratio σ is given by
Lateral contractional strain
σ=
Longitudinal extensional strain
Work done during stretching or Energy of the wire =
Energy density or work done per unit volume =

1
force x extension
2

1
stress x strain
2

Thrust is the normal force acting on a fluid surface.
Pressure at a point is thrust on unit area around the point.
Unit: Nm-2 (Pa). Dimensions: ML-1T-2
The pressure due to a liquid column of height h is given by
P=hρg
where ρ is density of liquid
Viscosity arises due to tendency of liquid layers to resist relative motion. The viscous force is given by
Newton’s formula
F=ηA

v
r

where η is coefficient of viscosity of the liquid, A area of cross section and

v
is velocity gradient.
r

Unit of coefficient of viscosity: Nsm-2. Dimensions: ML-1T-1
Poiseuille’s formula gives the rate of flow of liquid through a tube. The volume V of the liquid flowing
through the tube in t seconds is given by
π Pr 4
V
=
8ηl
t

where, P = pressure difference at the ends of the tube, r = radius of the tube, l = length of the tube, and η =

coefficient of viscosity of the liquid.
This formula is valid only when the motion is streamlined and the liquid has low viscosity.
The viscous force acting on a sphere of radius a, in a highly viscous medium of viscosity η is given by
Stoke’s law
F=6π η av
where, v is the terminal velocity of the sphere.
The terminal velocity of a sphere in terms of density of the sphere ρ and viscosity η
v=

2 a 2g
(ρ − σ )
9η

The critical velocity of a liquid is the velocity beyond which stremlined motion becomes turbulent. It is
given by
R eη
vc =
ρr
where, ρ is density of liquid, r is radius of the tube, η is viscosity of liquid and Re is Reynold’s number.
Bernoulli’s theorem states that the total energy of a moving fluid remains constant, in the absence of
friction. It can be written as
kinetic energy + potential energy + pressure energy = constant.
1 2
P
v + gh + = constant for unit mass of liquid
ρ
2
The velocity of efflux is the horizontal velocity with which the liquid flows through a narrow orifice in a
vessel. This is given by
v = 2gh

where h is the depth of the orifice from the liquid surface. This result is known as Torricelli’s theorem.
The equation of continuity states that for regular flow of a liquid
A 1v 1 = A 2v 2
where A1, A2 are areas of cross section of a pipe and v1, v2 respective velocities of liquids at these cross
sections.
Surface tension is the force per unit length on an imaginary line drawn on the surface of a liquid so that the
force is tangential to the surface and perpendicular to the line.
Unit: Nm-1 . Dimensions: MT-2. Surface tension is surface energy per unit area.


Physics at a Glance

The capillary rise of a liquid of density d, and angle of contact θ, surface tension S, in a tube of radius r
is given bny

11

2 S cos θ
rdg

h=

The excess pressure inside a spherical drop of radius r is
∆P =

2S
r

The excess pressure inside a spherical bubble of radius r is
∆P =

4S
r

The excess pressure inside a cylindrical drop of radius r is
∆P =

S
r

The normal force required in separating two glass plates containing a liquid film of thickness d and surface
tension S is

S
x area of contact between the film and glass plate.
d

The work done W against surface tension to increase the area is given by
W = increase in area x surface tension

9. HEAT AND THERMODYNAMICS
The linear expansivity α of a substance is defined as the increase in length per unit length per degree rise in
temperature
α=

∆l
l∆T

where ∆ l is the increase in length l original length, ∆T rise in temperature.
The areal expansivity β of a substance is defined as the increase in area per unit area per degree rise in

temperature
β=

∆A
A∆T

where ∆ A is the increase in area, A original area and ∆T rise in temperature.
The cubical (volume) expansivity γ of a substance is the increase in volume per unit volume per degree rise
in temperature.
γ=

∆V
V∆T

where ∆V is increase in volume, V original volume and ∆T rise in temperature.
It can be shown that
β = 2 α and γ = 3α approximately.
A liquid has two volume expansivities. They are apparent and real. If γ (real) is real expansivity and
γ(app) is apparent expansivity
γ (real) = γ (app) + γ (cont)
where γ (cont) is cubical expansivity of the container.
The density of a liquids d1 and d2 at two temperatures T1 and T2 respectively are related by the equation (T2
> T1)
d1 = d2 (1 + γ ∆T)
where ∆T is (T2-T1) and γ is the real volume expansivity of the liquid
Expansion of Gases

Pressure coefficient of a gas is given by
P − Po
β= t

Po x t
where Pt is pressure at t0C, Po is pressure at 00C at constant volume
Volume coefficient of a gas is given by
α=

Vt − Vo
Vo x t

where Vt is volume at t0 C and V0 volume 00C at constant pressure
For any two temperatures t1 and t2
α=

V2 - V1
V1t 2 - V2 t1

and β =

P2 - P1
P1t 2 - P2 t1


Physics for IIT-JEE Screening Test

12
Gas Laws

If P is the pressure of a gas, V its volume and T its temperature and ρ its density, then
PV = constant at constant T (Boyle’s law).
P
= constant at constant T is another form of Boyle’s law.

ρ

P
= constant at constant volume
T
V
= constant at constant pressure (Charles’ laws)
T
The ideal gas equation is a relation when all the above three parameters vary. For one mol of gas,
PV = RT and PV = nRT for n mol of gas
The heat energy required to raise the temperature of a body of mass ‘m’ and specific heat ‘c’ through ∆T is
given by
H = mc ∆T
A gas has three variables. They are pressure P, volume V, and temperature T. When a gas is heated, both
its pressure and volume change. For convenience, one of them is kept constant. So a gas has two specific
heats. They are specific heat at constant pressure (Cp) and specific heat at constant volume (CV)
Specific heat at constant volume CV is the heat required to raise the temperature of unit mass ( 1 mol)
through 1 K
Specific heat at constant pressure (Cp) is the heat required to raise the temperature of 1 mol of gas through 1
K at constant pressure..
Units : J mol-1 K-1. Dimensions: L2T-2K-1.
Cp exceeds CV by a factor equal to external work done by the gas
Mayer’s relation:
Cp- CV = R for 1 mol
R
= r for 1 kg.
Cp – CV =
m
(m is molecular mass and r is gas constant for 1 kg).
From the kinetic theory, root mean square velocity of gas molecules is

3P
ρ
where P is the pressure of the gas and ρ its density.
The rms velocity does not depend on pressure.
It is directly proportional to square root of temperature

Vrms =

3RT
( T- temperature, M- molar mass)
M
The average kinetic energy of a gas molecule per degree of freedom is given by

Crms =
E=

1
k BT
2

where k B is Boltzmann’s constant.
The average kinetic energy of one mole of monatomic gas is given by
3
E = RT
2
For one molecule, we divide the right hand side by N, the Avogadro number.
Mean free path is the average distance travelled by a gas molecule between two successive collisions.
Mean free path λ is given by Maxwell’s equation,
1
λ=

2π n σ 2
where σ is the diameter of the molecule and n is the number of molecules in unit volume.
Thermodynamics

It deals with study of interaction of heat and other forms of energy, especially mechanical energy.
Zeroeth law of thermodynamics states that ‘if two systems A and B are in thermal equilibrium
independently with a third system C, then A and B will be in thermal equilibrium with themselves.
The zeroeth law defines temperature. It says without a third system C (which could be thermometer), it
will not be possible to compare the temperature of two systems A and B.
First law of thermodynamics states that if we supply heat ∆ Q to a system, work done by the system is ∆W
and the increase in internal energy is ∆U, then


Physics at a Glance

13

∆ Q = ∆ U + ∆ W.
Internal energy of a thermodynamic system is the sum of potential and kinetic energies.
Work done by a thermodynamic system is the area of the P-V graph
Critical constants: Critical temperature (Tc) is that temperature above which a gas cannot be liquefied by
compression.
Critical pressure (Pc) is the pressure required to liquefy a gas at critical temperature.
Critical volume (Vc) is the volume of unit mass of gas at critical pressure and critical temperature. These
three are called critical constants.
Triple point is the temperature at which the three states of matter (solid, liquid and gas) remain in
equilibrium.
Triple point of water is 273.16 K at a pressure of 610 Pa (4.58 mm of Hg or 0.006 atmosphere).
Work done during an isothermal process at a temperature T, when a gas changes its volume from V1 to V2
is given by

⎡V ⎤
W = 2.3 RT log10 ⎢ 2 ⎥ for one mole. For n mol, we multiply the RHS by n
⎣ V1 ⎦
Work done during an adiabatic process
1
⎡
⎢= γ − 1 (P1V1 - P2 V2 )
for 1 mol. For n mol multiply the RHS by ‘n’.
W ⎢
⎢ R
=
(
T
T
)
1
2
⎢
⎣ γ −1
Thermodynamic processes:
Table 1.3 gives the thermodynamic processes, their definition, their equations and the application of the
first law of thermodynamics to them.
TABLE 1.3
Name of the
process

Definition of the
process

Equation to the

process

First law applied
to it
(∆ Q = ∆ U + ∆W)

Isothermal

Constant
temperature

PV= constant

∆ U = 0, ∆Q = ∆
W

Isobaric

Constant
pressure

P = constant
V/T = constant

∆W = p dV
∆Q = ∆ U + P dv

Isochoric

Constant volume

P/T = constant

dV = 0
∆Q=∆W

Total heat

PVγ = constant
TVγ-1 = constant
P1-γ Tγ = constant

∆Q=0
∆ U = -∆ W

Adiabatic

Second law of thermodynamics states that heat flows only from a body at higher temperature to that at
lower temperature. External work has to be done to transfer heat from a cold body to a hot body.
Transmission of heat:

Conduction: At thermal equilibrium (steady state) the heat conducted between the ends of a rod of crosssectional area A, length x and temperature T1 and T2, during time t is given by
λ A (T1 - T2 ) t
Q=
x
T − T2 dT
=
is known as the temperature
where λ is coefficient of thermal conductivity. The quantity 1
x

dx
gradient.
Units of thermal conductivity: Wm-1 K-1. Dimensions: MLT-3K-1
Convection is the mode of transmission of heat by actual motion of molecules. It is possible only in fluids.
Radiation is the transmission of heat as electromagnetic waves. Radiant heat travels with the speed of light.
It requires no medium.
Blackbody is one which absorbs all the heat falling on it and emits all radiation it has when heated top a
suitable temperature.
Blackbody radiation is the radiation emitted by a black body. The laws of Black body radiation are:
1. Wein’s displacement law states that λmT = constant, where λm is the wavelength corresponding to
maximum energy. The constant is known as Wein’s constant.


Physics for IIT-JEE Screening Test

14

2. Stefan’s law of radiation: The energy emitted by a black body per unit area per second is directly
proportional to the fourth power of its Kelvin temperature.
E = σ T4, where the constant σ is known as Stefan’s constant.
Units of Stefan’s constant: Wm-2K-4. Dimensions: MT-3K-4.
Relative emittance of a body is the ratio of heat emitted by the body per unit area in one second to that
emitted by a perfectly black body under identical conditions.
The radiation emitted from any body of relative emittance (emissivity) e, having an area A, at a temperature
T during a time t, into surroundings of temperature T0 is given by
E = σ Ate (T4-T04)
Heat Engines

A heat engine is a device, which converts heat energy into mechanical work. A heat engine absorbs a
quantity of heat Q1, from the source and rejects a heat Q2 to the sink, and thereby converts Q1-Q2 into

useful work. Its efficiency η is given by
Q - Q2
η= 1
Q1
The efficiency of an ideal heat engine (Carnot’s engine) working between two temperature T1 (source) and
T2 (sink) is given by
Q - Q2
T -T
η= 1
= 1 2
Q1
T1
A Carnot’s refrigerator is a reversible heat engine, which operates at the reverse cycle. It absorbs a
quantity of heat Q2 from the sink and rejects a quantity of heat Q1 to the source (surroundings). If W is the
external energy supplied, then Q1 = Q2 + W. The ratio

Q2
is known as coefficient of performance ω, and
W

is given by
ω=

Q2
Q2
=
W
Q1 − Q 2

=

T2
T1 − T2

The efficiency of refrigerator is the reciprocal of coefficient of performance. Hence efficiency η is given
by
η=

T1 - T2
T2

10. ELECTROSTATICS
Coulomb’s law

The electrostatic force of attraction or repulsion between two point charges q1, q2 at a distance r acts along
the line joining them and has a magnitude
Aq1 q 2
1 q1q 2
F=
=
,
2
4πε0 r
r2
where constant A =

1
4πε0

. Here ε0 is the permittivity of free space.

Value of ε0 = 8.85 x 10-12 Fm-1, value of A =

1
4πε0

= 9 x 109 Nm2 C-2 .

For any other medium ε0εr =ε should be written in the place of ε0, where εr is the relative permittivity of the
medium.
Electrostatic field at a point is the force acting on a unit charge placed at that point.
Two units of electrostatic field: N C-1 and Vm-1. Dimensions: MLT-3 I-1
Field due to a point charge q at a distance ris given by
E=

Aq
r2

Electrostatic potential at a point is the potential energy of a unit charge placed at the point.
Unit: JC-1 = volt (V). Dimensions: ML2T-3I-1
The electrostatic potential energy between two point charges q1, q2 at a distance r is
Aq1q 2
U=
r
An electric dipole is formed with two equal and opposite charges kept at a distance. The dipole moment is
the product of charge and distance, p = q x 2a. Dipole moment is a vector. It is directed from negative
charge to positive charge.


Physics at a Glance

15

Unit: C m. Dimensions: ITL
The field due to an electric dipole at a point along the axial line at a distance r from its centre is
A 2pr
A 2p
= 3
if a 2 << r 2
2
2 2
(r - a )
r
The field due to an electric dipole at a point on the equatorial line (perpendicular bisector) at a distance r
from its centre is
Ap
2

2 3/ 2

(r - a )

=

Ap
r3

if a 2 << r 2

The potential energy of a dipole in a uniform electric field E is

U = -p.E = -pE cos θ
where θ is the angle between field and axis of the dipole
The torque acting on a dipole in a uniform field is
τ=pxE
The magnitude of torque = pEsin θ. In a non-uniform field a dipole experiences both force and torque.
Potential difference between two points a and B in an electric field is VB – VA = line integral of electric
field. It is given by
BG

VB – VA = - ∫ E. dl
A

dV
dr
An equipotential surface is the locus of points having same potential. No work is done in carrying a charge
between any two points on the surface.

The electric field is the gradient of potential, i.e., E = −

Electric Flux

The flux through a given region in an electric field is φ = E. ds = E ds cos θ, where θ is the angle between
the electric field vector and area vector.
Gauss’s flux theorem states that the total normal flux crossing a closed surface in an electric field is
1
times the charge enclosed inside the surface.
ε0
q

∫ E. ds = ε

0

If there is more than one charge, the algebraic sum of charges should be taken on the right hand side.
By applying this theorem we can evaluate the electric field E due to the following:
1. A thin sheet of charge at a distance r
σ
(σ charge per unit area)
E=
2 ε0
2. A line of charge at a distance r ( A long thin needle)
λ
(λ charge per unit length)
E=
2πε 0 r
3. Two plane sheets of charge per unit area + σ1 and + σ2
E= 0, at a point inside
σ
σ
E = 1 + 2 at a point outside
2ε 0 2ε 0
4. Two plane sheets of charge per unit area + σ1 and - σ2
E= 0, at a point outside
σ + σ2
at a point inside.
E= 1
2ε 0
Further if σ1 = σ 2 = σ , this will be
E=

σ
. This is the equation for the field between the parallel plate condenser
ε0

Capacitance is the ratio of charge q to potential V.
q
C=
V
Unit: farad (F). Dimensions: M-1L-2T4I2


Physics for IIT-JEE Screening Test

16

The capacitance of a parallel condenser is
ε0ε r A
C=
d
where A is area of the plates, d separation of the plates, εr relative permittivity (dielectric constant) of the
medium between the plates.
Capacitance of a conducting sphere of radius ‘r’ is given by
C = 4πε0 r
Capacitance of a spherical condenser is given by
4πε 0 ε r r1 r2
C=
r1 + r2
where r1 , r2 are the radii of inner and outer concentric spheres the condenser is made of.
The capacitance of a cylindrical condenser made from two concentric cylinders of radii r1 and r2 and of
length L is

2πε 0 ε r L
C=
2.3 x log10 (r2 / r1 )
where εr is relative permittivity of the medium between the cylinders.
Capacitance of a parallel plate condenser with two dielectric media of thickness t1 and t2 and dielectric
constants k1 and k2 respectively is
ε 0 k 1k 2 A
C=
k1t 2 + k 2 t1
Three capacitors of capacitance C1, C2, C3 joined in series have an effective value C given by the equation
1
1
1
1
=
+
+
C C1 C 2 C 3

When ‘n’ identical capacitors each of capacitance C are joined in parallel, the effective value of capacitance
is nC. When n identical capacitors are in series will have an effective value

C
.
n

The energy of a charged conductor of capacitance C, charge q and potential V is given by
E=

1

CV 2 in terms of C and V
2

1 q2
in terms of C and q
2 C
1
E = qV in terms of q and V
2

E=

11. ELECTRO DYNAMICS
Ohm’s Law

If states that the current flowing through a conductor I is directly proportional to the potential difference V
between the ends.
V
=R
I ∝ V or V ∝ I ⇒
I
where the constant R is called the resistance of the conductor
Unit of R: ohm (Ω). Dimensions of R : ML2T-3I-1
The resistance of the conductor of length and cross-sectional area A is given by
R=

ρL
A

where ρ is called resistivity (specific resistance). Unit: Ω m.

Three resistors R1, R2, R3 in series have an effective value R, given by
R = R1 + R2 + R3
Two resistors R1, R2 in parallel have an effective value R given by
1
1
1
R1R 2
=
+
or R =
R R1 R 2
R1 + R 2

Three resistors R1, R2, R3 in parallel have an effective value R given by
R1R 2 R 3
1
1
1
1
or R =
=
+
+
R R1 R 2 R 3
R1R 2 + R 2 R 3 + R1R 3


Physics at a Glance

17

‘n’ identical resistors each of value R in series have an effective value nR. ‘n’ identical resistors each of
value R in parallel have an effective value

R
n

Conductivity of a conductor σ is given by
ne 2 τ
σ=
m
where n is the number of free electrons per unit volume in the conductor, , e charge of electron, m its mass
and τ relaxation time.
The current I flowing through a conductor
I=nevA
where n is number of free electrons per unit volume, e charge, v drift velocity and A cross-sectional area.
The current density j is given by
current nevA
= nev
j =
=
A
area
The resistance of a conductor changes with temperature. if Rt , R0 are the resistances at t0 C and 00C
respectively, the temperature coefficient α is given by
R −R0
α= t
R0t
More generally, If R1 and R2 are the resistances at t10 C and t20 C respectively, the temperature coefficient is
R 2 − R1

α=
R 1t 2 − R 2 t1
For metals α is positive and for semi-conductors α is negative.
To convert a galvanometer into an ammeter, we connect a low resistor S called shunt in parallel.
IgG
S=
I − Ig
where Ig is the current flowing through galvanometer, I total current, G resistance of galvanometer.
To convert a galvanometer to a voltmeter we connect a high resistor R in series given by
V
R=
-G
Ig
where V is the voltage to be measured and G resistance of galvanometer.
Kirchoff’s Laws

1. The algebraic sum of currents meeting at any junction is zero.
2. In any closed mesh, the algebraic sum of product of current and resistance is equal to the algebraic sum
of emf acting on the mesh.
Kirchoff’s first law is law of conservation of charge, while the second is law of conservation of energy.
The condition of balance of a Wheatstone’s bridge is P/Q = R/S, where P and R are resistors in the left side
up and down respectively while Q and S resistors on the right side up and down of the bridge.
The balancing condition of a meter bridge is
Re sis tan ce on left
Balancing length on left
=
Re sis tan ce on right Balancing length on right
Potentiometer: It is an instrument to measure electromotive force. If E1 is the e.m.f. of a cell which
balances l1 m of the wire, ‘i’ current through potentiometer, ‘r’ resistance per metre of the wire, then E1 = i r
l1. The ratio of emfs of two cells balancing lengths l1 and l2 is given by

E1
l
= 1
E2
l2
The internal resistance of a cell using potentiometer is given by
R ( L − l)
B=
l
where L is balancing length in open circuit and l balancing length with R ohm in parallel with the cell.
Magnetic Field of Charges

A charge moving with a velocity v in a uniform magnetic field of flux density B will experience a force
F=qvxB
Magnitude of the force is qvB sin θ, where θ is the angle between velocity and field vectors. The direction
of the force is perpendicular to the velocity and field vector.
The path of a charged particle in a uniform magnetic field is a circle, when the field is perpendicular to the
velocity. The radius of the circle is given by


Physics for IIT-JEE Screening Test

18

mv
.
qB
If the field makes an angle, the path will be helix.
The force acting on a conductor of length l carrying a current I in a uniform magnetic field of strength B is
F=IlxB

Magnitude of this force = IlB sin θ is an angle between field vector and length.
The direction of this force is given by Fleming’s left hand rule. If three fingers of left hand (fore finger,
middle finger and thumb) are held at right angles, the fore finger is pointed in the direction of field, the
middle finger in the direction of the current, then the thumb will give the direction of the force or that of
motion.
The torque acting on a current carrying coil in a magnetic field is given by
τ = nIA x B
where n is the number of turns of the coil, I current, A area and B magnetic field. Magnitude of the torque
is nIAB sin θ, where θ is the angle between area and field vectors. Torque is maximum when plane of the
coil is parallel to the field (θ = 900). Torque is zero when plane of the coil is perpendicular to the field (θ =
0)
The magnetic moment of a current loop of area A and carrying current I,
µ = current x area of the loop = IA
The current through a moving coil galvanometer of n turns, of area A, in a uniform magnetic field B is
⎡ c ⎤
I= ⎢
⎥θ
⎣ nAB ⎦
r=

where θ is the deflection of the coil in the field, c is couple for twisting the suspension wire through one
radian.
Magnetic Field of Currents

Magnetic field due to a current element of length dl, carrying a current I, at a point distant r from the
element is given by Biot-Savart law,
µ I d l sin θ
A I d l sin θ
dB = 0
=

2
4π
r2
r
µ0
where θ is angle between the line joining the element and direction of current. Constant A =
,
4π
where µ0 is called permeability of free space. It has a value 4π x 10-7.
Units: henry / metre (Hm-1). For any other medium, we have to replace µ0 by µ0 µr , where µr is the relative
permeability of the medium.
The direction of the magnetic field will be perpendicular to the plane of current element I dl and the
radius vector r.
Using Biot -Savart Law we can evaluate field due to conductors
µ I
B = 0 (sin θ1 + sin θ 2 )
4π a
where θ1 and θ2 are the angles between the line joining the point to the ends of conductor and perpendicular
through the point to the conductor.
Field due to an infinitely long straight conductor carrying current i is
µ I
B= 0
2π a
Field due to a circular coil of n turns of radius a, carrying current I at a distance x along the axis from its
centre is
µ 0 na 2 I
B=
2( a 2 + x 2 ) 3 / 2
Field at the centre of the coil (x = 0)
µ nI

B= 0
2a
Field due to a long solenoid having n turns/ unit length carrying a current Ii at its centre is
B = µ0 nI.
Field due to a long solenoid at a point on the axis and at one of its ends is ,
µ nI
B= 0 .
2


Physics at a Glance

19

Field due to a toroid carrying a current I is
B = µ0 nI
Here n is the number of turns per unit length or total number of turns / 2πr.
Force between two long parallel wires carrying currents I1, I2 and separated by a distance a in free space or
air
µ I I
F = 0 1 2 Nm −1
2πa
The force is attractive if currents are in the same direction and repulsive currents are in the opposite
direction.
Magnetic Properties of Materials

Intensity of magnetisation M is the induced magnetic moment per unit volume .
Magnetic susceptibility is the ratio of intensity of magnetisation, M to magnetising field, H.
M
K=

H
‘H’ here is measured in ampere per metre
Permeability is the ratio of magnetic induction B to magnetising field, H.
B
µ=
H
Relative permeability is the ratio of final magnetic induction through the specimen to induction through the
same region in the absence of the specimen.
B
µr =
B0
The relation between relative permeability and susceptibility is µ r = 1 + K
Terrestrial (Earth’s) Magnetism

The magnetic elements of earth are (i) Dip (ii) Declination and (iii) Horizontal intensity
Declination is the angle between geographic meridian and magnetic meridian.
Dip or magnetic inclination is the angle between total intensity of earth’s magnetic field and the horizontal.
Horizontal intensity of earth’s magnetic field is the resolved part of total intensity in the horizontal
direction in magnetic meridian.
If H is the horizontal intensity, V is the vertical intensity, I is the total intensity, δ the angle of dip, the
relation among these are
H = I cos δ,
V = I sin δ
I=

V2 + H2 ,

V
H
Tangent law: If two magnetic fields F and H at right angles are applied simultaneously to a magnetic

F
needle free to rotate in the plane of the fields, it will settle to an angle θ with H such that
= tan θ.
H
⎡ 2aH ⎤
Tangent galvanometer: The current passing through tangent galvanometer I is given by I = ⎢
⎥ tan θ ,
⎣µ0 N ⎦
2aH
where
= K, is known as the reduction factor. Thus I is proportional to tan θ. Here ‘a’ is radius of
µ0 N
the coil, ‘n’ number of turns and ‘H’ horizontal component of earth’s field.
Vibration magnetometer: The period of oscillation of a magnet in a uniform field is given by
I
I
, i.e. T 2 = 4π 2
T = 2π
mB
mB
1
1
We find T2 ∝
when H is constant and T2 ∝
when m is constant. Here m is moment of the magnet,
m
B
I is its moment of inertia, B is the field in which the magnet oscillates.
If T1 is the period when two magnets are placed one over the other with their like poles pointing in the
same direction, T2 the period when one of them is reversed, the ratio of magnetic moments is

tan δ =

m1
T 2 + T2 2
= 12
m2
T2 − T1 2


20

Physics for IIT-JEE Screening Test

12. ELECTROMAGNETIC INDUCTION
The phenomenon of electromagnetic induction is the production of electric current from changing magnetic
field.
Faraday’s law of electromagnetic induction states that the induced emf and induced current are proportional
to the rate of change of magnetic flux. If φ is the magnetic flux and e is the induced emf, this law can be
written as
d
dφ
e∝
(φ) ∝
dt
dt
Lenz’s law: The induced emf and the induced current are always in a direction opposite to the rate of
change of flux. Combining these two laws we can write
dφ
and the induced current ‘I’ is given by
e=dt

1 dφ
I=R dt
Rule to find the direction of induced current using Lenz’s law: If the north pole of the magnet is moved
towards a coil, for an observer standing on the other side of the coil, the induced current will be in the
clock-wise direction.
Induced emf when a conductor of length l, moves with a velocity v in a magnetic field B, with angle θ
between field and length, is given by
e = Blv sin θ
Induced emf when a conducting rod is rotated in a uniform field, with plane of rotation perpendicular to the
field is
1
e = Bl2 ω
2
where l is the length, ω angular velocity, B field
General equation for induced emf for a coil is
d
e=
(BAN cosθ)
dt
where, B is the flux density of magnetic field, A is the area of cross-section which intercepts the flux, N is
the number of turns, θ is angle between area vector A and field vector B.
The direction of induced emf can be found by Fleming’s right hand rule. If three fingers of right hand (fore
finger, middle finger, and thumb) are held in mutually perpendicular directions with fore finger in the
direction of the field and thumb in the direction of the motion of the conductor, the middle finger will give
the direction of induced current.
Mutual induction: Whenever current through a coil changes, an emf is induced in the neighboring coil.
This phenomenon is called mutual induction. The induced emf is
dI
dI
e = -M

, where
is rate of change of current in the coil and M is coefficient of mutual
dt
dt
induction.
Units of M: henry (H)
Self Induction: Whenever current through one coil changes, an emf is induced in itself. This phenomenon
is called self-induction. The induced emf is
dI
e = -L
dt
where L is coefficient of self-induction. L is also measured in the same unit as M. i.e. henry (H).
Mutual inductance between two coils (primary and secondary), primary having a total number of turns N1
and length l, secondary having a total number of turns N2 is
µ µ AN1 N 2
M= 0 r
l
Here A is area of cross-section of the primary.
The self inductance of a coil having N turns, length l, area of cross section A is given by
µ µ N2A
L= 0 r
l


Physics at a Glance

21

13. ALTERNATING CURRENT (AC) AND AC CIRCUITS
Alternating current is produced when a coil rotates with constant angular velocity in a uniform magnetic

field. The induced emf when a coil of N turns, area A, rotates with angular velocity ω in a field B is given
by
E =[NAB ω] sin ωt = E0 sin ωt
where E0 = NAB ω. E0 is called the peak value. It is the maximum value of emf
The average value of a.c. (voltage and current) over a cycle is zero. The RMS value of a.c. during one
cycle is given by
E
Erms = 0 = 0.7 E 0
2
I0
Irms=
= 0.7 I 0
2
AC Circuits

Reactance is the resistance offered by a component other than the resistor. It is denoted by X.
The resistance offered by an inductor in an a.c. circuit is called inductive reactance XL. The resistance
offered by capacitor to an a.c. is called capacitative reactance Xc,
XL = ωL
1
Xc =
.
ωC
Here ω = 2πf, where ‘f’ is the frequency of a.c.
The impedance of an a.c. circuit is the vector sum of resistance and reactance. The letter Z denotes
impedance. It is given by
Z= R 2 + X 2
The frequency of oscillation of an L.C circuit (resonant circuit) is
1
f=

2π LC
where L is inductance and C capacitance
The power in an a.c. circuit is given by
P = Erms Irms cosφ
where cos φ is called power factor.
R resis tan ce
=
.
Cos φ =
Z impedance
If cos φ = 0 ⇒ φ = 900. The current flowing is wattless current. Average power consumed is zero. This
happens in a circuit containing pure inductor or pure capacitor.
Q-factor and resonance curve: Q-factor or quality factor of a circuit is the ratio of inductive reactance to
the resistance of the circuit at resonance. At resonance, both reactances will be equal in magnitude. Hence
Lω
.
Q=
R
Resonance curve is a graph drawn between frequency and current. At resonance, current (hence absorption
1
of energy) is maximum. The frequencies for which current falls to
of maximum current (that is half
2
the maximum power) are called half power frequencies (f2 and f1). The difference f2-f1 is called the
bandwidth. Q-factor can also be defined as the ratio of resonant frequency to bandwidth,
fr
Q=
f 2 − f1
Eddy currents: When a solid block of conducting material is rotated in a magnetic field, induced currents
are produced due to electromagnetic induction. These currents flow in closed paths on the surface of the

block. Such currents are called Eddy currents or Foucault currents.
Transformer: A transformer is a device to increase or decrease AC voltage. It works on the principle of
electromagnetic induction (mutual induction). Whenever there is a change in the current through one coil,
an emf is induced in the neighboring coil.
If Ep and Es are the voltages across the primary and the secondary, then from the theory of the transformer,
it can be shown that


Physics for IIT-JEE Screening Test

22

Np

E
= P
NS ES
where Np and Ns are the number of turns in the primary and secondary respectively. It should be
remembered that a transformer does not increase electric power. Even for an ideal transformer
Ep Ip = Es Is.
Main sources of losses in a transformer are Copper loss (Joule heating), Eddy current loss, Magnetic flux
linkage loss, Hysterisis loss.
1
Energy stored in an inductor is LI2, where L is the inductance and I is the current.
2
This energy is stored in the form of magnetic field.

14 THERMOELECTRICITY AND CHEMICAL EFFECTS
Seebeck Effect

When two different metal wiores are jolined to form two junctions and kept at two different temperatures, it
is found that an emf is set up in the circuit. This phenomenon is called Seebeck effect. The emf thus
produced is called the thermo emf and current, the thermoelectric current and the arrangement,
thermocouple.
The graph of thermo emf against temperature is a parabola whose equation can be written as
e = a (θ - θ0) + b (θ - θ0)2
where a and b are constants for a given pair of metals, θ0 is the cold junction temperature and θ is hot
junction temperature.
Neutral Temperature and inversion Temperature

The temperature at which thermo emf becomes maximum is known as neutral temperature θN. The
temperature at which thermo emf reverses in direction is known as inversion temperature θi. The relation
between the two is,
θ + θi
θN = 0
2
where θ0 is cold junction temperature. θN is constant, while θi varies with θ0.
Peltier coefficient: The heat energy absorbed or evolved due to passage of current at a junction is
proportional to current and time, i.e. charge H ∝ q. That is
H=πq
where the constant π is known as Peltier coefficient. Unit of π is volt.
Thomson effect is the production of emf when there is a temperature gradient in a conductor. Thomson
coefficient is the potential difference between two points of a conductor which has a temperature gradient
of one unit. Its unit is VK-1
Faraday’s Laws of Electrolysis

Law 1: Thisstates that the mass of substance liberated during an electrolysis is directly proportional to the
quantity of charge passing through it. If m is the mass of the substance liberated due to q coulomb of
charge then, m ∝ q or m ∝ It, where I is current and t is time.
m = eq

The constant e is called electro-chemical equivalent
Law 2: This states that if same quantity of electricity is passed through different electrolytes, the quantity of
substances liberated will be proportional to the respective equivalent weights. If m1, m2, m3 are the masses
of substances liberated by same quantity of electricity and z1, z2, z3 their respective equivalent weights, then
the law states that,
m1 m 2 m 3
m
=
=
. i.e.
= constant
z1
z2
z3
z
z
is one Faraday = 96500 C or 9650 emu
e
It is charge required to liberate one gram- equivalent of a substance.

The constant


Physics at a Glance

23

15. ELECTROMAGNETIC WAVES
The speed of light in free space c =

1
µ0ε0

. In other media light travels with a speed

1
µε

Displacement

current is the current, which actually does not flow but produces the effect of current, i.e., it produces a
magnetic field. This is the type of current that flows between condenser plates.
Electromagnetic spectrum consists of a family of radiation characterised by constant velocity through free
space, 3 x 108 ms-1. It starts from γ rays of short wavelength of about 10-12 m and extends to radio waves
of wavelength of the order of km.
Long Distance Communication and Radio Waves

Two kinds of waves used for communication are (i) Sky waves and (ii) Ground waves.
A sky wave is reflected by the ionosphere. A ground wave travels along the surface of earth.
The range of a TV signal transmitted as ground wave by an antenna of height h is given by
x = 2Rh
where R is the radius of earth.

16. WAVE OPTICS
Interference

It is a non-uniform distribution of light energy due to two or more sources. Interference is produced by two
coherent sources. Two sources are coherent if they produce waves of same wavelength, same amplitude,
same phase or a constant phase difference.
Conditions of Interference: Two waves arriving at same point produce brightness (constructive

interference) if their path difference is equal to nλ, where n is an integer = 0,1,2,3,....etc. They produce
λ
darkness (destructive interference) when path difference is equal to (2n + 1) where n= 0,1,2,3, etc.
2
The bandwidth of interference bands in Young’s double slit experiment is given by
Dλ
β=
d
where D is the distance of the source from the viewing microscope, d separation of two coherent sources
(slits) and λ wavelength of light.
The angular width of a band is given by
β λ
= .
D d
The resultant intensity of two waves of phase differenceφ at any points is given by
⎡φ⎤
I = I0 cos2 ⎢⎣ 2 ⎥⎦
where I0 is the maximum intensity.
A thin film of thickness t and refractive index µ appears dark by reflection when viewed at an angle of
refraction r if
2µ t cos r = nλ (n = 1, 2, 3 etc)
The minimum thickness (n=1) of a film which appears dark by reflection at normal incidence
(r = 00) is
2µ t = λ
The minimum thickness of a film, which appears bright under normal incidence of monochromatic light of
wavelength λ is
λ
2µt=
2
The band width β of fringes produced by an air wedge formed with two glass plates separated by diameter

(d) of a wire, is given by
λ
d
tan θ =
=
2β l
where ‘l’ is the length of the glass plate forming the wedge, θ angle of the wedge.
Newton’s rings are circular interference fringes formed with a glass plate and a convex lens enclosing an
air film of varying thickness. The diameter of n th dark ring dn is given by
d n2 = 4Rnλ (n = 0, 1, 2, 3,....etc)
Thus, diameter of dark rings will be proportional to square root of natural numbers.


24

Physics for IIT-JEE Screening Test

Diameter of nth bright ring is
λ
(n = 0, 1, 2, 3....etc)
d 2n = 4R (2n + 1)
2
If we introduce a medium of refractive index µ between a convex lens and a glass plate, we have to
substitute µdn2 in the LHS of the above equation.
Diffraction

It is the deviation of light from a rectilinear path or bending of light round the corners of an obstacle and
spreading of light into a geometrical shadow. For diffraction by a single slit (Fraunhoffer diffraction) of
width a, the n th minima appears at an angle θn given by
a sin θn = nλ (n = 1,2,3....)

The central maximum is at θ = 0 . The n th maximum appears at angle θn, given by
λ
a sin θn = (2n + 1) (n = 1,2,3,....)
2
2λ
.
The angular width of central maximum is 2θ given by 2θ =
a
A diffraction grating is made by a number of parallel slits ruled on a glass plate using a fine diamond point.
The n th maximum appears at an angle θn given by equation
sin θn = Nn λ
where N is the number of slits in unit length of the grating. If a is the width of the slit and b distance
1
between the slits, then a+b is called grating element. N =
.
a+b
Polarisation

It is the property of one-sidedness, where light vibration is only in one direction in the transverse plane.
The orientation of the electric vector will be only in one direction.
Polarisation can be produced by simple reflection. It light is reflected by a transparent medium at an angle
i, the whole reflected light will be polarised if
tan i = n
Here i is called the polarising angle and n is the refractive index. This result is known as Brewster’s law.
Double refraction is the formation of two refracted images of one object. It is shown by selected class of
crystals, i.e. quartz, calcite and to some degree ice.
A quarter wave plate produces a path difference of quarter wave-length (λ/4) or a phase difference of π / 2
between 0 and E rays. It is made by a doubly refracting crystal. The thickness of a quarter-wave plate is
given by
λ

(ne – n0 ) t =
4
where ne, n0 are refractive indices of O (ordinary) and E (extraordinary) ray.
A half-wave plate produces a path difference of half wavelength (λ / 2) or a phase difference of π between
O and E rays. The thickness of a half wave plate t is given by
λ
(ne – no) t =
2

17 RAY OPTICS
Velocity of Light by Michelson’s Method

Equation for velocity of light in Michelson’s rotating mirror method is
c = 2Nnd,
where N is number of faces of the mirror, n number of revolution per second of the mirror, d is the distance
between concave mirrors (distance travelled by light one way).
Spherical Mirrors

r
2
Sign conventions: (i) All distances are measured from the pole of spherical mirror. (ii) Distances of real
objects and real images are taken as positive while those of virtual objects and virtual images are taken as
negative.
The mirror formula or law of distances is
1 1 1
= + .
f u v
This is valid only for paraxial rays.

The relation between focal length (f) and radius of curvature of the mirror (r) is f =

Physics at a Glance

25

Here u is distance of the object, v distance of the image, f the focal length of the mirror..
Magnification:

There are two magnifications. The lateral magnification is the one produced when the object is
perpendicular to the principal axis. It is given by
v
in terms of u and v.
m=
u
f
m=
in terms of u and f
u−f
v−f
m=
in terms of v and f.
f
The longitudinal magnification is the one produced when the object is parallel to the principal axis. It is
given by
m=

v2

2

⎛ v−f ⎞
⎛ f ⎞
=⎜
⎟
⎟ =⎜
2
−
u
f
u
⎝ f ⎠
⎝
⎠

2

Refraction

Snell’s law of refraction states that
sin i
= constant
sin r
where i is the angle of incidence and r the angle of refraction. This constant is known as the refractive
index (n) of the refracting (second) medium with respect to the first (incident) medium.
If ng and nw are refractive indices of glass and water respectively, refractive index of glass with respect to
water
ng
ngw =
nw

When an object in a denser medium is viewed from a rarer medium, the object appears elevated. The
refractive index of denser medium
Actual depth of the object
n=
Apparent depth
The distance through which the object appears to be elevated is given by
(n − 1)d
x=
n
where d is the actual depth of the object.
If a ray of light passes from a denser to a rarer medium, the ray totally reflects, when the angle of incidence
exceeds a certain value C. This angle is known as critical angle of the denser medium. The relation
between refractive index n and critical angle is
1
n=
sin C
The angle at which an under water swimmer sees the setting sun is equal to the critical angle of water C =
⎡3⎤
⎡4⎤
sin-1 ⎢⎣ 4 ⎥⎦ = nearly 480 with vertical (assuming refractive index of water as ⎢⎣ 3 ⎥⎦ )
A fish under water sees the outside world within a cone of a semi vertical angle equal to the critical angle
of water C = 480
Radius of the cone at water surface (r )
3
= tan C =
Depth of fish (h )
7
Refraction through a prism: If A is the angle of prism, D is the angle of minimum deviation, the
refractive index of prism ‘n’ is given by
( A + D)

A
/ sin
n = sin
2
2
If i1 is the angle of incidence in the first face of the prism, r1 the angle of refraction in the first face, r2 the
angle of incidence in the second face, i2 the angle of refraction in the second face (or angle of emergence)
and d is the angle of deviation
r 1 + r2 = A
i1 + i2 = A+d
For a small angled prism, the deviation d can be written as
d = (n-1)A and the refractive index n is given by