1. Attempt all of the following

Discuss the magnetic elements of the Earth,

with the help of a neat sketch. Give the

relationships among these elements.


What is fault plane solution of an earthquake

Using the 'Beach Ball' image show the fault

plane solutions for Normal fault, Thrust

fault, Vertical strike-slip fault, and

Vertical dip-slip fault.


Write down the expressions for gradient of a

scalar field, and divergence and curl of a

vector field, in Cartesian coordinate

system.

What is Singular Value Decomposition of

a matrix How is it used to obtain the inverse

of a square matrix


2. Write down the expressions of Laplace equation

in Cartesian, cylindrical and spherical

coordinate systems.

Describe an iterative linear inversion method

for quasi-linear problems and discuss the

problem of trapping of the solution in local

minima.

Discuss the magnetic field of the Earth in

terms of internal and external causes. Describe the

theory, with the help of a neat sketch, that

explains 90% of the observed magnetic field of

the Earth.




3. Describe the minimum norm solution of an

problem using Lagrange under determined

multiplier.


Show that the minimum time-path between

points A and B for the following ray geometry

gives the same result as Snell's law




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With the help of a neat sketch, describe how the

time of occurrence of an earthquake and the

distance travelled by the seismic wave be

determined, using Wadati diagram.

4. What do you understand by 'Monte Carlo' and

'Directed Monte Carlo' methods Describe the

important steps of Monte Carlo method of

inversion.

Discuss the mode of propagation of EM field

for low 50kHz) and high 50 k.Hz)

frequencies, On the basis of it, define the

quasi-static approximation and quasi-static:

region for low frequency EAf field.


Discuss the geophysics evidences of 'Seafloor

spreading'.




5. Expand cosh z cosh 6 in a Taylor series in

powers of 6). What is the radius of

convergence of this power series


Find the wavelength and propagation speed of

radio waves of 1 MHz in copper and compare

your answers with those in vacuum (give:n that

conductivity of copper is 6 x 10^7 SI units,

8·85 x 10^-12 SI units. µo x SI

units; also assumed µo)



1 kg of water at 0°C is brought in contact with a

heat reservoir at 100°C.


Calculate the change in entropy of water

when it attains the temperature of the

reservoir.


Obtain the entropy change of both the

reservoir and the universe.


Draw a neat block diagram of a 3-layer model

for a satellite navigation system.

Discuss the working principle of the above

navigation system.




6.

A periodic function with a period of is given by

x^2 for x
by


Find out the Fourier series expansion and use

your result to find the sum of the series

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Write down Maxwell's equations for a plane

electromagnetic wave propagating in free space.

How do these equations differ from those in an

isotropic conducting medium

Show that the plane electromagnetic wave


E follows the equation


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What is the origin of day-time attenuation of

the high frequency radio waves Write a

brief note on this specific ionospheric layer.



7. Write down Simpson's rule for numerical
integration of

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by dividing the interval in four equal


parts. Compare the numerical value with the

exact result and give your comments.


What is atmospheric plasma Define electron

and ion plasma frequencies in terms of the

material parameters. Show that the plasma

effects on electromagnetic wave propagation in

the ionosphere is mainly due to the electrons.



Using the relation

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Here, λ c/v and is the black body

emissivity and h is Planck's constant.

Using the variable x show that the

value of for which is maximum is given

by 0


8. Define a group and show that the set of all

2 x 2 unitary matrices with determinant 1 is a

group. Find the number of independent real

parameters required to specify a group element




The first excited state of He-atom lies

19·82 eV above the ground state. If this excited

state is 3-fold degenerate while the ground

state is non-degenerate, find the relative

populations of the first excited state and the

ground state for He-gas in thermal equilibrium
at 1000 K.




Show that the refractive index of the

ionospheric medium can be approximated as

n


with N as electron density and f is the linear

frequency in kHz.

What will happen, if the electron plasma

frequency is greater than the wave

frequency?