UNIVERSITY OF ENGINEERING & MANAGEMENT, Jaipur
Syllabus Covered - Engineering Physics (PH-201)
Module -1
Physical significances of grad, div, curl. Line integral, surface integral, volume integral- physical examples in the context of electricity
and magnetism and statements of Stokes theorem and Gauss theorem [No Proof]. Expression of grad, div, curl and Laplacian in
Spherical and Cylindrical co-ordinates.
Module -2
Electricity 2.1 Coulombs law in vector form. Electrostatic field and its curl. Gauss’s law in integral form and conversion to differential
form . Electrostatic potential and field, Poisson’s Eqn. Laplace’s eqn (Application to Cartesian, Spherically and Cylindrically symmetric
systems - effective 1D problems) Electric current, drift velocity, current density, continuity equation, steady current. 5L 2.2
Dielectrics-concept of polarization, the relation D=ε0E+P, Polarizability. Electronic polarization and polarization in monoatomic and
polyatomic gases.
Module -3
Magnetostatics & Time Varying Field: 3. Lorentz force, force on a small current element placed in a magnetic field. Biot-Savart law
and its applications, divergence of magnetic field, vector potential, Ampere’s law in integral form and conversion to differential
form. Faraday’s law of electro-magnetic induction in integral form and conversion to differential form.
Module -4
4.1 Concept of displacement current Maxwell’s field equations, Maxwell’s wave equation and its solution for free space. E.M. wave
in a charge free conducting media, Skin depth, physical significance of Skin Depth, E.M. energy flow, & Poynting Vector
Module -5
5.1 Generalized coordinates, Lagrange’s Equation of motion and Lagrangian(no derivation), generalized force potential, momenta and
energy. Hamilton’s Equation of motion (no derivation) and Hamiltonian. Properties of Hamilton. (physical problems of 1-D motion)
5.2 Concept of probability and probability density, operators, Commutator. Formulation of quantum mechanics and Basic
postulates, Operator correspondence, Time dependent Schrödinger’s equation, formulation of time independent Schrödinger’s
equation by method of separation of variables, Physical interpretation of wave function ψ (normalization and probability
interpretation), Expectation values, Application of Schrödinger equation - Particle in an infinite square well potential (1-D and 3-D
potential well), Discussion on degenerate levels.
Module -6
Statistical Mechanics: 3.1 Concept of energy levels and energy states. Microstates, macrostate and thermodynamic probability,
equilibrium macrostate. MB, FD, BE statistics (No deduction necessary), fermions, bosons (definitions in terms of spin, examples),
physical significance and application, classical limits of quantum statistics Fermi distribution at zero & non-zero temperature,
Calculation of Fermi level in metals, also total energy at absolute zero of temperature and total number of particles, Bose-Einstein
statistics - Planck’s law of blackbody radiation..
UNIVERSITY OF ENGINEERING & MANAGEMENT, Jaipur
Question Bank - Engineering Physics (PH-201)
Module -1
1
a. If vectors A and B are irrotational then show that vector A  B is solenoidal.
b. Find the unit vector perpendicular to 4 x 2  3 y 2  2 z 2  50 at the point (2,3,5).
2+3
2
a. If A  B  A  B , then show that A and B are perpendicular to each other.
2+3
b. Prove that .v  0 , where  is a constant vector , r is the position vector and v    r .
3
a. Evaluate  
2
b. Prove that  ln r 
4
2+3
r
.
r
1
.
r2
a. Calculate the work done in moving a particle in a force field given by F  2 xyiˆ  4 zjˆ  8 ykˆ
along the curve x=t2+1, y=t2, z=t3 from t=0 to t=1.
2+3
b. Find the directional derivative of ( x, y , z )  xy 2 z  4 x2 z at (-2,1,1) in the direction 2iˆ  ˆj  2kˆ .
5
a. Calculate the work done in moving a particle in a force field given by F  xyiˆ  yzjˆ  xzkˆ along
2+3
the curve r  tiˆ  t 2 ˆj  t 3kˆ where t varies from -1 to 1.
b. In what direction from the point (-2,1,1) is the directional derivative ( x, y , z )  2 xz  y 2 is
6
maximum? What is the magnitude of this maximum?
a. Find a unit vector normal to the surface x2-y2-z2=5 at the point (-2,1,1).
b. Find the angle between the surfaces x2+y2+z2=5 and x2+y2-z=7 at the point (-2,1,1).
7
a. If B   A , show that
b. Find
8
9
10
11
12
 B.d S  0 for any closed surface.
2+3
2+3
 r.d S  0 where the surface S encloses a volume V and r is the position vector of any
point on this surface.
𝑞
𝑞
An electric Field intensity is given as 𝐸⃗ = 100 cos 𝜃 𝑟3 𝑟̂ + 50 sin 𝜃 𝑟3 𝜃̂ then
(i)
Find |𝐸| at r =2, 𝜃 = 600, ∅ = 200
(ii)
Unit vector in Cartesian coordinates in direction of 𝐸⃗ .
Express the vector 𝐴 = 3xy î– 5 (x+z) 𝑘̂ in cylindrical coordinates and evaluate it at (2, 600, 3).
Given Point P(x=3, y=2, z=-1) and Q(r=2, 𝜃=300, ∅ = 600), Evaluate
(i)
Spherical Coordinates of P
(ii)
Cartesian Coordinates of Q
𝜋𝑥
For a scalar function ∅ = (sin 2 sin
increase of ∅ at the point (1,1,1).
𝜋𝑦
)
3
𝑒 −𝑧 calculate the magnitude and direction of maximum rate of
A vector field is given as W  4 x 2 yiˆ  (7 x  2 z ) ˆj  (4 xy  2 z 2 )kˆ
(i)
What is the magnitude of the field at point (2,-3,4)?
(ii)
At what point on z –axis is the magnitude of W equal to unity?
2+3
5
2+3
5
5
13
14
15
a. Find the constant m such that the vector   ( x  3 y )iˆ  ( y  2 z ) ˆj  ( x  mz )kˆ is solenoidal.
b. Give physical significance of gradient, divergence and curl.
a. Give statement and physical significance of Gauss-divergence theorem.
b. Give statement and physical significance of Stoke’s theorem.
Give conditions for (i) solenoidal fields (ii) irrotational fields (iii) incompressible liquids (iv) solenoidal but
not irrotational fields.
2+3
2+3
5
Module -2
1
(i)
(ii)
(iii)
(iv)
(v)
State and explain Coulomb’s law in electrostatics.
Express it mathematically with meaning of each symbol for two point charges.
Does it depend on the medium property? If yes, then answer, how?
What is the most important requirement for the validity of Coulomb’s law?
Show that gravitational force can be neglected when compared with Coulomb’s force.
5
2
(i)
(ii)
Define electric field E and electric potential V at a point and how they are related.
Why electric field intensity is called conservative field?
5
Show that  E  0 .
What is the unit of electric field intensity and electric potential?
In a place the electric potential is same everywhere. What is your understanding about the
electric field intensity in that place?
Find the potential and hence the electric field due to (a) a uniformly charged rod (b) uniformly charged
ring (c) uniformly charged disc.
(i)
What is electric flux? Define it.
(ii)
What is the total flux across a closed surface due to a charge kept outside the surface?
(iii)
State and explain Gauss’ law in electrostatics. What are its limitations?
(iv)
Derive Coulomb’s law from Gauss’ law.
(v)
Write down the differential and integral form of Gauss’s law.
Using Gauss’ law find the electric field intensity outside, inside and on the surface of (a) uniformly
charged sphere (b) hollow charged sphere (c) uniformly charged cylinder (d) hollow charged cylinder.
Also show the graphical representation.
a. Applying Gauss’ law determine the electric field intensity due to (a) long thin wire of uniform
charge distribution (b) infinitely plane sheet of charge.
b. The amount of net charge enclose by a closed surface is known, but there is no idea about the
distribution of charges. In this case, can Gauss’ law be applied to determine the electric field
intensity at any point of closed surface? Explain.
a. Extend Gauss’ law to Poisson’s equation. When does it reduce to Laplace’s equation?
b. Write down Laplace’s equation in Cartesian co-ordinate system. Two infinite parallel plates at
z=0 and z=a are maintained at potentials V0 and Va respectively. Obtain the variation of
potential and field between the plates.
a. Write down Laplace’s equation in Spherical co-ordinate system and find the solution for
spherical capacitor considering the variation of potential along radial direction.
b. Write down Laplace’s equation in cylindrical co-ordinate system and find the solution for
cylindrical capacitor considering the variation of potential along radial direction.
(i)
What do you mean by dielectric and dielectric constant of a material?
(ii)
What is meant by polarization of dielectric? How the dielectrics are categorized on the basis of
polarization? What are polar and non-polar dielectrics?
(iii)
Define the following terms: (a) dipole moment (b) electric susceptibility (c) atomic
polarizability.
(iv)
What is the unit of dipole moment, polarization, atomic polarizability?
(v)
Explain the phenomena of polarization of dielectric medium and show that K=1+χ, where the
(iii)
(iv)
(v)
3
4
5
6
7
8
9
5
5
5
2+3
2+3
2+3
5
symbols have their usual meanings.
10
(i)
(ii)
(iii)
(iv)
(v)
11
12
ε (ε −1)
Show that the electronic polarizability α = 0 r .
N
Show that D =ε0E + P.
Prove that electric polarization is equal to the surface charge density of polarization.
Show that when the dielectric is placed in an electric field, the field within the dielectric
becomes weaker than the original field.
What happens when a non-polar molecule is placed in an electric field?
a. Two point charges Q and q are placed at a distance x and x/2 respectively from a third charge
4q. All the three charges are on the same straight line. Calculate Q in terms of q such that the
net force on q is zero.
b. Four equal charges +q are placed at the corner of a square. Find the point charge at the centre
of the square so that the system will remain in equilibrium.
c. An amount of charge Q is divided into two particles. Find the charge on each particle so that the
effective force between them will be maximum.
a. Check whether the field E= 4yi – 2xj + k is conservative.
2+2+1
1+2+2
b. If E= q/(4πε0r2) r then show that E is solenoidal.
c. If the potential in the region of space near the point (-2m, 4m, 6m) is V= 80x2 + 60y2 volt, what
are the three component of electric field at that point?
13
a. If the electric field on a region is E  4iˆ  6 ˆj  7kˆ find the electric flux through the surface
area of 75 square units in XY plane.
b. S1 and S2 are two hollow concentric spheres enclosing charges Q and 2Q respectively. What is
the ratio of electric flux through inner surface S1 and outer surface S2?
2+3
14
a. For positive x, y and z, let V = 40 xyz c/m3. Calculate the total charge for the regions defined by
3+2
(i) 0  x, y, z  2 and (ii) x  0, y  0;0  2 x  3 y  10;0  z  2
b. The electric potential V(x) in a region along the x-axis varies with distance x (in meter) according
to the relation V(x) = 4x2. Calculate the force experienced by 1 mC charge placed at point x= 1m.
15
a. Given the potential V 
1+2+2
10
sin  cos  . Find the work done in moving a charge -10  C from a
2
r
point A (1,300,1200) to B (4,900,600).
b. Find the electric flux through each face of a unit cube due to a charge q coulomb placed at (i)
centre (ii) one of the vertices.
c. A spherical shell of inner radius r1 and outer radius r2 is uniformly charged with charge density ρ.
Calculate the electric field at a distance r from the centre of the spherical shell for (a) r>r2 (b)
r1<r<r2 (c) r<r1.
16
a. In cylindrical coordinates (  ,  , z ) , electric flux density is given by D  z  cos 2  zˆ C/m2. 3+2
  
,3  and the total charge enclosed by the cylinder of radius
 4 
Calculate the charge density at  1,
1 meter with 2  z  2 meter.
b. A capacitor uses a dielectric material of dielectric constant 8. It has an effective surface area of
0.036 m2 with a capacitance of 6 μF. Calculate the field strength and dipole moment per unit
17
18
volume if a potential difference of 15 V exists across the capacitor.
a. A dielectric material contains 2 × 109 polar molecules/m3 each of dipole moment 1.8 × 10–27 cm. 3+2
Assuming that all of the dipoles are aligned towards electric field E = 105 V/m. Find the
polarization, electric susceptibility and the relative permittivity.
b. The dielectric constant of helium at 0°C is 1.0000684. If the gas contains 2.7x10 25 atoms/m3,
find the radius of the electron cloud.
3+2
a. Verify that potential function Vx, y  3x2  2 y 2 satisfies Laplace's equation or not. Also find the
charge density.
b. The electric field in a region is given as E= kr3 r . Prove that charge contained within a spherical
surface of radius a is 4πε0ka5.
Module -3
1
2
3
4
5
6
7
a. Starting from the definition of current and current density, derive the equation of continuity.
b. Write the condition of steady state current. Show that Ampere’s law implies that the current is
in the steady state?
c. Compare the Lorentz electric force and Lorentz magnetic force.
a. State and explain Biot-Savart’s law.
b. Using the law, calculate the magnetic field of induction due to (a) a long straight wire (b) circular
current carrying coil (c) long current carrying solenoid.
a. Show that the field at the end of a long current carrying solenoid is half that of at the centre.
b. Express Biot-Savart’s law in terms of current density and hence show that the magnetic field is
solenoidal.
a. State Ampere’s circuital law.
b. Deduce differential form of Ampere’s circuital law and hence prove that the static magnetic
field is not conservative.
c. Applying Ampere’s law find the magnetic field due to (a) long straight current carrying solid and
hollow cylinder. Draw the necessary diagram explaining the variation of magnetic field with
distance.
d. From Ampere’s circuital law find the magnetic field due to a long solenoid carrying a current I
and number of turns per unit length being n.
e. Find the force per unit length of a current carrying conductor placed in a uniform magnetic
field. Hence find the force between straight conductors carrying current (a) in same direction
(b) opposite direction.
a. What do you mean by magnetic scalar potential and magnetic vector potential?
b. If the curl of magnetic field in current free space is zero, find the expression of magnetic scalar
potential.
c. Derive the expression of magnetic vector potential in terms of current density J.
d. Write down Poisson’s equation and Laplace’s equation for vector potential.
a. Find the magnetic field at the center of (a) an equilateral triangular loop and (b) square loop
both having side a and carrying current I.
b. Two wires carrying current in the same direction of 5,000 A and 10,000 A are placed with their
axis 5 cm apart. Calculate the force between them. Justify whether the force is of repulsion or
attraction type.
a. A circular current carrying coil has a radius R. Show that the distance from the centre of the coil,
on the axis, where B will be (1/8) of its value at the centre of the coil is (√3r).
b. A straight wire 0.5m long carries a current of 100A and lies at right angles to a uniform field of
1.5T. Find the mechanical force on the conductor when it lies in a position such that it is inclined
at an angle of 30° to the direction of field.
c. A horizontal overhead power line carries a current of 50 A in west to east direction. What is the
2+2+1
2+3
2+3
2x5=10
5
3+2
2+2+1
8
9
magnitude and direction of the magnetic field 1.5m below the line?
a. If a charged particle of charge 0.4 C is moving with a velocity 4î − ĵ + 2k̂ m.s-1 through an 3+2
electric field E  10iˆ  10kˆ , and magnetic field of induction B  2iˆ  16 ˆj  6kˆ , then find (i)
the force experienced by the particle in electric field (ii) the force experienced in magnetic field
and (iii) Lorentz force.
b. A small rectangular wire loop (3cmx2cm) carries a current of 5 A in anti-clockwise directions. A
long straight wire carrying a current of 10 A is placed in the plane of the loop parallel to longer
side of it. The nearest longer side of the loop is 5 cm away from the long wire. Find the net force
on the loop. Is it attracted or repelled by the long wire?
a. Two wire loops ABCDA formed by joining two semi circular wires of radii R1 and R2 carries a
3+2
current I as shown in figure. Find out the magnetic field at centre O.
b. The wire shown in figure carries a current I. What will be the magnitude and direction of
magnetic field at the centre?
10
11
12
13
14
a. Consider a coaxial cable which consists of an inner wire of radius a surrounded by an outer shell 3+2
of inner and outer radii b and c respectively. The inner wire carries a current i and outer shell
carries equal current but in opposite direction. Find the magnetic field at a distance r, where (i)
r<a (b) a<r<b (iii) b<r<c (iv) r>c.
b. If the vector potential A= (2z+5)i + (3x-2)j + (4x-1)k, find the magnetic field.
a. A particle of charge q moves with a velocity v = ai in a magnetic field B=bj+ck where, a,b,c are 2+3
constants. Find the magnetic force experienced by the particle.
b. There is a crossed field with uniform electric field E and a uniform magnetic field B
perpendicular to each other. A positively charged particle having charge q enters the region of
crossed fields perpendicular to both E and B with a velocity v. Show that it will traverse the
region un-deflected if its speed satisfy the relation v=E/B.
a. Two circular coils each of radius 6cm and 60 turns are separated by a distance of 16 cm along their 3+2
common axis. Find the strength of the magnetic field of induction at a point midway between them on
their common axis when a current of 0.1 A is passing through them.
b. Calculate the magnetic field intensity along the axis if a solenoid carrying a current of 3 A. The length of
the solenoid is 1.5 m and the total number of turns is 1500.
a. A solenoid has 4 layers of 1200 turns each. Its length and mean radius are 3 m and 0.25 m respectively.
Find (i) The magnetic field at its center, if a current of 2.5 A flows through it (ii) magnetic flux at its center
for a cross section of the solenoid.
b. Calculate the magnetic field intensity just outside and inside of a hollow cylinder of radius 4 cm carrying
current 50 A. Also find the magnetic field of induction B.
a. A short conductor of length 5 cm is placed parallel to a conductor of length 1.5 m. Both conductors
carrying a current of 3 A and 2 A respectively in the same direction. Find the nature and magnitude of the
force experienced by the long conductor for their separation of 3 cm.
If the vector potential A  (10 x  y  z ) ˆj at any position, then find the (i) magnetic field at that
position (ii) magnetic field at the point (1,1,1).
a. A square loop of wire of edge ‘a’ carries a current I. show that the value of the magnetic induction 𝐵⃗ at
b.
15
2
the center of the loop is given by B 
2
3+2
3+2
2
2 2 0 I
where  0 is the permeability of free space.
a
b. A solenoid has length 3 m and mean radius 0.25 m. Find the magnetic flux density at its center when a
current of 2 A flows through its 3 layers of 1500 turns each.
3+2
Module -4
1
2
3
4
5
6
7
8
a. State Faraday’s law of electromagnetic induction and express it in differential and integral form.
b. Define displacement current. Prove that it is equal to conduction current.
c. Show that Ampere’s law is insufficient for time varying field. Hence find the modified form of
Ampere’s law.
a. Write down Maxwell’s equation with statements in differential form and explain physical
significance of each.
b. From differential form of Maxwell’s equations obtain the integral form.
Write down Maxwell’s equation for the following cases (a) free space or vacuum (b) good conducting
medium (c) conducting medium (d) dielectric medium (v) static field.
a. Write down Maxwell’s equation in differential form in free space. From these equations derive
the wave equation for an electromagnetic wave. What is the velocity of this wave?
b. Prove that electromagnetic wave is transverse in nature.
a. Write down Maxwell’s equation in charge free conducting medium. Find solution of these in
conducting medium.
b. Show that the electromagnetic wave attenuates as it propagates through a conducting medium.
a. What is Poynting vector? What is its physical significance? Give statement of Poynting theorem.
b. A parallel plate capacitor with circular plates of 10 cm radius separated by 5 mm is being
charged by an external source. The charging current is 0.2 A. Find the (i) the rate of change of
potential difference between the plates and (ii) obtain the displacement current.
a. Show that for frequency <=109 Hz, a sample of silicon will act like a good conductor. For silicon,

 12 and  =2 mho/cm. Also calculate the penetration depth for this sample at
assume
0
frequency 106 Hz.
b. A metal bar slides without friction on two parallel conducting rails at distance r apart. A resistor
R is connected across the rails and a uniform magnetic field B, pointing into this plane fills the
entire region. If the bar moves to the right at a constant speed v then what is the current in the
resistor?
a. A rectangular loop of sides 8 cm and 2 cm having resistance of 1.6  is placed in a magnetic
field of 0.3 Tesla directed normal to the loop. The magnetic field is gradually reduced at the rate
of 0.02 T/s. Find the induced current.

 1 . Calculate the highest frequencies for

which the earth can be considered a good conductor if <<1 means less than 0.1. assume  =5 x 10-3
mho/m and   10 0 .
a. Calculate the value of Poynting vector for a 60W lamp at a distance 0.5 m from it.
b. Calculate the value of Poynting vector at the surface of the sum if the power radiated by the sum is 3.8 x
1026 W and its radius is 7 x 108m.
2+1+2
3+2
5
3+2
2+3
3+2
2+3
2+3
b. The earth is considered to be a good conductor when
9
10
11
12
2+3
a. A plane e-m wave whose electric field is given by E= 24 sin(πx107t – 0.5πz) V/m travels in a 3+2
perfect dielectric. Find its velocity and the corresponding magnetic field.
b. Calculate the skin depth for radio waves of 3 m wavelength (in free space) in copper, the electrical
conductivity of which is 6 x 107 S/m.
a. Define skin depth and give mathematical expression for the same explaining meaning of each term
involved.
b. Find the magnetic field B and Poynting vector P of electromagnetic waves in free space if the components
of electric fields are Ex=Ey=0 and Ez  E0 cos kx sin t .
2+3
a. Find the skin depth  at a frequency 1.6 MHz in aluminium where 𝜎 = 38.2 x 106 mho.m-1 and 𝜇 = 4𝜋 x 2+3
10-7 henry.m-1.
b. An ac voltage source is connected across the two plates of an ideal parallel plate capacitor. If the applied
13
ac voltage V = V0 sin 𝜔t, then verify that the displacement current in the ideal capacitor is equal to the
conduction current through the wire.
a. If the average distance between the sun and the earth is 1.5 x 1011 m, show that the average solar energy
incident on the earth (called solar constant) is 2 cal.cm-2.min-1.
b. An electromagnetic wave is propagating through a medium in such a manner that electric vector of the
2+3
2 E
wave satisfies the differential equation   
. How much energy will be absorbed by the
t 2
2
medium in 20 seconds?
Module -5
1 1. (a) What are the limitations of Newtonian Mechanics?
(b) What do you mean by constraints, constrained motion and force of constraint?
(c) Define scleronomic, rheonomic, holonomic and non-holonomic constraint with example.
2
a. What is meant by degrees of freedom of a dynamical system? Discuss it for a system of N
particle with k number of constraints.
b. Specify the nature of constraint in the following cases:
(i) a simple pendulum with rigid support (ii) a pendulum with variable length (iii) a particle is
constrained to move on the surface of a sphere (iv) a bead is constrained to move on a circular
wire (v) motion of a body on an inclined plane under the influence of gravity
(vi) motion of a
body on an ellipsoid under the influence of gravity
(vii) a sphere rolling down from the top
of a fixed sphere (viii) rigid body (ix) any deformable body (x) motion of gas molecules in a
cubical container.
3
a. Determine the number of degrees of freedom for the following cases:
(i) a particle moving on the circumference of a circle (ii) a particle moving on the surface of a
sphere (iii) five particles moving freely in space (iv) two particles connected by a rigid rod
moving freely in a plane (v) dumbbell moving in a space (vi) NH3 molecule.
b. What do you mean by virtual displacement. Show that the work done by constraint force is zero.
4
a. State the principle of virtual work and explain it.
b. What are advantages of using generalized coordinates and hence give expression for
generalized kinetic energy.
5
a. What are the generalized coordinates needed to specify the motion of the following:
(i) a particle constrained to move on the circle in the XY plane (ii) a particle constrained to move
on an ellipse in the XY plane (iii) a particle constrained to move on the surface of a sphere (iv) a
bead sliding on a wire in the form of a cycloid (v) two masses of a double pendulum constrained
to move in a plane.
b. What should be the dimension of Qjdqj .
6 2. Obtain the expression of generalized velocity, generalized momentum, and generalized forces.
a. A spherical ball of very small radius is suspended through a point on the ceiling using an inextensible
7
b.
8
1+2+2
2+3
3+2
2+3
4+1
2+1+2
3+2
string of some length. Ball is then given some displacement in horizontal direction. Calculate the
Lagrange’s equation of motion for the ball and also find frequency of oscillation.
Calculate the Lagrange’s equations of motion for a particle of mass m falling freely under gravity near the
surface of earth.
a. A particle of mass m moves in a force field of potential V. The kinetic energy and the potential 3+2
energy of the system is T 


1
m r 2  r 2 2  r 2 sin 2  2 , V= -V(r,θ,φ). Write the equation of
2
motion using the Lagrangian method and Hamiltonian method.
b. Find equations of Hamiltonian for a particle falling freely under gravity.
9
Use Lagrange’s equation to find the equation of motion of the following systems:
5
(a) free particle (b) mass spring system which oscillates on a frictionless horizontal plane (c) simple
pendulum (d) compound pendulum (e) particle moving under the influence of central force (f) a
system of two masses connected by an inextensible string passing over a small smooth pulley (g)
LC circuit (h) freely falling body under the influence of gravity.
10
10
11
12
Write down the Hamiltonian and derive Hamiltonian equation of motion for the systems given in Q. No
9.
a. Discuss the advantages of Lagrangian formulation over Newtonian formulation of mechanics.
b. What are the advantages of Hamiltonian formulation over the Lagrangian formulation?
c. Write Lagrange’s and Hamilton’s equations of motion.
a. What is the physical significance of the wave function Ψ? What are the conditions of any
acceptable wave function Ψ?
b. What is probability density? How is wave function related to the probability of finding a particle
at any point in space at a given time?
c. What do you mean by normalized wave function?
a. Clarify whether the following wave functions are acceptable or not
(a)
Ψ(x) = e.
(b) Ψ(x) = 1/(1+x)
b. The wave function of a particle moving in one dimensional box of length a is given by
 ( x) 
13
15
A  a2  x2 
a
2+2+1
2+2+1
2+3
for 0<x<a
=0
outside
Find A that will normalize  ( x ) .
a. A particle is moving along x-axis and the wave function associated with the particle is given by
Ψ(x) = ax
for 0 <x < a
And
=0
everywhere.
Find the probability that the particle lies between x=0.35 and x=0.75.
b. The wave function of a certain particle is 
2
 A cos x
for


2
particle to be in x = 0 and x =
5

 x

2+3
. Find (i) A (ii) probability for
2
.
4
14
15
a. Write down the position and momentum operators. Then check whether [x, px] is commutative 2+3
or non-commutative.
 
 , 
b. Find the energy and momentum operators. Find the value  x t  .
2
3+2
 2 d2 
x
2
a. Show that  ( x)  cxe
is an eigen function of the operator  x  2  . Find the eigen
dx 

value.
𝑑
b. Obtain the expression for eigen function of the momentum operator 𝑝̂𝑥 = -ih𝑑𝑥 corresponding to
an eigen value px.
16
17
pˆ x  i
d
corresponding to an eigen value px.
dx
a. Derive Schrödinger time dependent wave equation using De-Broglie wave equation for a free 3+2
particle.
b. Obtain Schrödinger time independent wave equation using Schrödinger time dependent wave
equation.
A particle of mass m is confined within x=0 and x=L.
5
a. Write down the Schrodinger equation for such a system.
b. Solve the equation to find out the normalized wave functions.
c. Hence show that the energy eigen value is quantized.
18
19
20
d. Show that the eigen functions corresponding to different eigen values are orthogonal.
a. Find the expectation value of x, x2, p and p2 in the ground state.
2+3
b. A system has two energy eigen states ε0 and 3ε0. Ψ1 and Ψ2 are the corresponding normalized
wave function. At an instant the system is in a superpose state Ψ = C1Ψ1 + C2Ψ2 & C1= 1/√2.
(i) Find the value of C2 if Ψ is normalized.
(ii) What is the probability that an energy measurement would yield a value 3ε0.
And (iii) Find out expectation value of energy.
Write down Schrodinger’s equation for a particle of mass m in a rectangular box with perfectly rigid 5
walls with edges a, b and c. Use separation of variable technique to calculate the eigen function and
eigen value of the particle. Calculate the wave length of a photon that must be absorbed for a transition
from lowest to 2nd excited state.
a. What is degeneracy of energy states? Explain.
b. Prove that 1st excited state of a free particle which is in a box of length L has 3-fold degeneracy.
3+2
Module -6
1
2
3
4
5
6
7
8
What do you mean by (a) macro state (b) microstate (c) ensemble (d) micro canonical ensemble (e)
canonical ensemble (f) grand canonical ensemble (g) thermodynamic probability (h) phase space.
a. Calculate the total no. of macrostate and microstate of a system consisting of ε, 2ε and 3ε
energy states with total energy 4ε and two distinguishable particles.
b. Calculate the total no. of macrostate and microstate of a system consisting of ε, 2ε and 3ε
energy states with total energy 4ε and two indistinguishable particles
a. Distribute three particles in two different states according to (i) MB (ii) BE (iii) FD statistics.
b. Three distinguishable particles each of which can be accommodated in energy states E, 2E, 3E.
4E with total energy 6E. Find all the possible number of distributions. Also find total microstates
in each case.
a. Write down basic postulates of MB, BE and FD statistics. Compare MB, BE and FD statistics
according to their particle nature, no. of particles in a state, energy distribution and spin.
b. Give examples of boltzons, bosons and fermions.
a. Discuss Fermi distribution function with graphical representation at zero and non zero
temperature.
b. Show that at absolute zero temperature the no of fermions per unit volume within energy
1
range ε to ε + dε is proportional to 𝜀 ⁄2 .
a. Show that the average electron energy is equal to 3/5 th of the Fermi energy at absolute zero.
b. Calculate the value of Average velocity, Fermi velocity and Fermi temperature.
a. Use BE statistics to obtain Planck’s radiation formula for black body radiation.
b. Calculate the Fermi energy in copper. Consider density of copper as 8.94 x 103 kg/m3 with
atomic mass 63.5 amu.
a. The Fermi energy for sodium at T=0K is 3.1 eV. Find its value for aluminium. Given that the free
electron density in aluminium is approximately 7 times that in sodium.
b. If the Fermi energy of a metal at thermal equilibrium is 15 eV , then find the average energy of
the electron.
9
5
3+2
2+3
3+2
2+3
2+3
3+2
2+3
3+2
a. Find the electron concentration of silver atom with atomic weight 108 and number of free
electron per atom as one and Fermi energy 4.5 eV at 0 K.
b. Calculate Fermi energy at 0K of metallic silver containing one free electron per atom. The
density and atomic weight of silver is 10.5 g/cm3 and 108 g/mol respectively.
10
2+3
a. There are about 25 x 1028 free electrons/m3 in sodium. Calculate its Fermi energy, Fermi velocity
and Fermi temperature.
b. A system with non-degenerate single particle state with 0,1,2,3 energy units. Three particles are
to be distributed in three states so that the total energy of the system is 3 units. Find the
number of microstates if particles obey (i) MB statistics (ii) FD statistics.
11
3+2
a. Show that at T=0, the average energy E of an electron in a metal is given by E 
3
E f (0)
5
where Ef(0) is Fermi energy at absolute zero.
b. If the Fermi energy of any metal is 10 eV. What is the corresponding classical temperature?