Section-A
1. Answer all the five parts below J2X5=6Q
Two identical relativistic particles of rest mass m and kinetic energy T collide head-on. What is the kinetic energy T of one in the rest frame of the other?
Define a conservative field. Determine if the field given below is conservative in nature
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where c is a constant.
Calculate the moment of inertia of a solid cone of mass height vertical half-angle (alfa) and radius of its base about an axis passing through its vertex and parallal to its base.
A resistance R and a lossless capacitor is charged to potential V0,and the switch is closed at t=0. Prove that the energy stored in the capacitor is equal to the energy dissipated in the resistor
A rigid body is spinning with an angular velocity of 4 rad/s about an axis parallel to the direction (4j passing through the point A with OA 2i 3j where O is the origin of the coordinate system. Find the magnitude and direction of the linear velocity of the body at point P
with OP =4i -2j+k.

2. The volume between two concentric conducting spherical surfaces of radii a and b is filled with an inhomogeneous dielectric with E where c is a constant and r is the radial coordinate. Acharge is placed on the inner surface, while the outer surface is grounded. Determine-

D in the region a r
capacitance of the device;
polarization charge density in the region a r
surface polarization charge densities at r a and r b.

3. A particle is moving in a central force field on an orbit given by r ke(alfa theta), where k and alfa are positive constants, r is the radial distance and theta is the polar angle.
Find the force law for the central force field.
Find theta(t).
Find the total energy.

4. What is a displacement current? Prove that lines of conduction current plus displacement current are continuous. 15

consider a uniform half-sphere of radius R and mass M. The half-spere is supported by a frictionless horizontal plane as shown in the figure. The half-spere lies in the region z 0.
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Find the centre of mass of the half-sphere.
Assume E 0 inside a perfect conductor. Elaborate on any other four electrostatic properties that arise from this property.
A projectile of mass M explodes, while in flight, into three fragments. One fragment of mass m1=M/2 travels in the original direction of of the projectile. Another fragment of mass m2=M/6 travels in the opposite direction and the third fragment of mass m3=M/3 comes to rest. The energy released in the explotion, is 5 times the kinetic energy of the projectile at explosion. What are the velocities of the fragments?

Section-8
5. Answer all the five parts below 12x5=60
The motion of a damped mechanical oscillator is represented by
mx+alfax+betax=O
where alfa and beta are constants. The oscillator is critically damped. The system is given an impulse at x 0 and t resulting in an initial velocity v. After how much time the system
experiences maximum displacement?
Show that a travelling wave on the string, clamped on both the ends, undergoes a phase change of pie. Hence obtain the time-independent form of the wave equation representing a standing wave on the string.
The separation between the slits is 0·5 mm in Young's double-slit experiment. The interference pattern observed on a screen placed 5 m away reveals the location of the first maximum which is 6 mm from the centre of the pattern. Calculate the wavelength of light and separation between second and third bright fringes.
Show that the Helmholtz free energy of a system never increases in any isothermal-isochoric transformation.

Consider non-equilibrium situation for a system in which the population inversion has been achieved. Explain that such a system can be treated as if it has negative absolute temperature.
6. Distinguish between Fresnel and Fraunhofer classes of diffraction. Show that the area of each Fresnel half-period zone is same.
A diffraction grating of width 5 cm with slits of width 10 4 cm separated by a distance of 2 x 10 cm is illuminated by light of wavelength 550 nm. What will be the width of the principal maximum in the diffraction pattern? Would there be any missing orders? 20
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Hence show that the heat capacity Cp of an ideal gas is independent of pressure P.

7. Derive an expression for intermodal dispersion for a multimodal step-index fibre. 20

A pulse of 600 nm and delta^ propagates through a fibre which has a material dispersion coefficient of 50 ps per km per nm at 600nm. Calculate the pulse broadening in traversing a 10 km length of the fibre. If the pulse width at the input of the fibre is 12 ns, what will be the pulse width at the output of the fibre?
Show that both Fermi-Dirac and Bose-Einstein distributions reduce under certain condition in a form which gives the total number of particles as
<img src='./qimages/212-7c.jpg'>
where A is a constant and beta k8T. Further show that this expression is just the same as that obtained from the Maxwellian speed distribution.

8. use matrix method to obtain an expression for the focal length of a coaxial combination of two thin lenses
having focal lengths f1 and f2 separated by distance d.
Calculate the minimum thickness of a quartz plate which would behave as a quarter-wave plate for wavelength of light, 6000 A. The refractive indices for ordinary and extraordinary rays are u0 1.544 and µe l.553. 15

(c)The Einstein theory of specific heat of solids gives the expression
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Mention Einstein's assumptions in deriving it. Also obtain low- and high-temperature limiting expressions for it.
Give schematic plot of Cv/3Nk8 verses T/thetaE and comment on the validity of expressions in in comparison with experiments.