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Examination, 2024 | SGSE-P-GPH
GEO-PHYSICS
Paper - I
Time Allowed : Three Hours Maximum Marks : 200
Question Paper Specific Instructions
Please read each of the following instructions carefully before attempting
questions :
There are TEN questions divided in Two sections.
Candidate has to attempt SIX questions in all.
Question Nos. 1 and 6 are compulsory.
Out of the remaining EIGHT questions, FOUR questions are to be attempted choosing
TWO from each Section.
The number of marks carried by a question/part is indicated against it.
Neat sketches are to be drawn to illustrate answers, wherever required. They shall be
drawn in the space provided for answering the question itself.
Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary, and indicate the same clearly.
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a
question shall be counted even if attempted partly.
Any page or portion of the page left blank in the Question-cum-Answer (QCA) Booklet must
be clearly struck off.
Answers must be written in ENGLISH only.
SGSE-P-GPH 1
Ql. (a)
(b)
(c)
(d)
Q2. (a)
(b)
(c)
SGSE-P-GPH
SECTION A
Derive Adams—Williamson equation and its application in Geophysics.
Also give variation of Body-wave velocity (a) in the Earth’s interior
according to Earth Model IASP91.
State Fermat’s principle and derive Snell’s law using the same principle
for geometry of incident and refracted ray on horizontal interface.
Write down the condition under which the magnetic dipole is presumed
to be a point or isolated pole. Draw the magnetic effects in terms of H, Z
and F profiles due to an isolated pole for I = 45°, where I is the field
inclination.
The diagonal elements of a covariance matrix computed for a linearized
inverse problem having model parameters m,, Mo, M3, My, Ms are 49,
15, 3, 200, 40 respectively. Compute the standard deviation in the
estimation of model parameter m,.
(i) Give Kepler’s laws of planetary motion relating the empirical laws
as expressions of fundamental physical laws.
(ii) Define the following terms as parameters of an elliptical orbit :
(1) Aphelion and Perihelion
(2) Perigee and Apogee
Explain, briefly, Newton’s law of gravitation. If a satellite is orbiting the
Earth 250 km above the surface, what acceleration due to gravity does it
experience ? Given that the radius of the Earth is 6-38 x 10° m and the
mass of the Earth is 5-98 x 1024 kg.
Gi) Find the general formula for inverse of 2 x 2 square matrix.
(ii) | Compute the inverse of the 3 x 3 square matrix given as
2. 4,-.$
A=] 1 -2 -2}.
-3 3 2
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Q3. (a)
(b)
(c)
Q4. (a)
(b)
(c)
Q5. (a)
SGSE-P-GPH
What forces are related to Isostasy ? A large area of continent consists of
30 km of crust with an average density of 2:8 g/cc and over 90 km thick
material with density of 3-1 g/cc. It is covered with a 1-6 km thick layer
of ice (density 0-9 g/cc) and is in isostatic equilibrium. Then the ice
melts. After equilibrium has been regained, by how much has the rock
surface of the continent changed ? (Density of asthenosphere is 3-2 g/cc).
Describe the distortion of current flow at the plane interface using
suitable sketch. Explain image theory approach in the case of electrical
prospecting. Compute the value of reflection coefficient, k, for
P, = 10 ohm-m and p, = 40 ohm-m, where p, and Po are the resistivities
of the two media.
Discuss the following in seismic wave propagation :
(i) Seismic Wave Attenuation
Gi) Seismic Wave Dispersion
What causes Geomagnetism? Give the definition of geomagnetic
elements and the relationship of Cartesian coordinates (Xp, Yo, Ze) and
spherical polar (F., D, I) sets of Earth’s magnetic elements.
(i) Explain the role of eigenvalues and eigenvectors to solve the
inverse problem.
(ii) Find out the eigenvalues and eigenvectors of a 2 x 2 square matrix
ar
Define ray parameter for spherical earth. Using ray parameter, derive
the expression for travel time and epicentral distance.
(i) Draw focal mechanism for the following :
(1) Normal-faulting earthquake
(2) Thrust-faulting earthquake
Gi) Describe the following seismic phases :
(1) PKJKP
(2) PKiKP
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10
to bo
(b)
(c)
SGSE-P-GPH
(11) Describe the Gutenberg—Richter relationship in seismology and its
applications.
Consider a thin wire carrying a current. Show the relation between
electric and magnetic fields using suitable sketch. Explain Maxwell’s
equations for the electromagnetic fields.
Consider a linear system A, = b,
3241 ಸತ್ತ -1
whereA=/]-1 3 -1 |, 204 0 | 7
1 7
0) Can we use Steepest Descent (SD) method for solving the above
linear system ?
Gi) If yes, compute three iterations by SD method starting with
20 = [0, 0, 0]7.
10
Q6. (a)
(b)
(c)
(d)
Q7. (a)
SGSE-P-GPH
SECTION B
Find the solution of the following differential equation
2
ee -~y=0, y=yit),
when y satisfies the initial conditions y(0) = 2 and * (0) = a, where a is
a positive constant. Then find the value of a so that the solution
approaches zero as t > ~, 10
Suppose we have some charge and current configuration in space which,
~
at time t, produces an electric field E ( t) and a magnetic field
a>
B (7; t). In the next instant (t + dt), the charges move around a bit.
Show that the work done per unit time, =
Pg by the electromagnetic
forces acting on the charges is equal to the rate of decrease in the energy
stored in the fields contained in the volume V enclosed by the surface S,
léss the energy that flowed out through the surface S. 10
Starting from Maxwell equations prove :
(i) Coulomb’s Law 5
(ii) Continuity Equation 5
Find the exact altitude of a GPS satellite that has an orbital period
equal to precisely one-half of a sidereal day. Use a value of mean earth
radius r, = 6378-14 km and sidereal day length of 23 hours 56 minutes
4-1 seconds. 10
Calculate and determine the Laurent series for the function
1
owas 8
valid for the region {z: | z | <5}. 10
(b)
(c)
Q8. (a)
(b)
(c)
Q9. (a)
(b)
(c)
SGSE-P-GPH
> >
Write down the expressions for the real electric (E ) and magnetic (B )
fields for a monochromatic plane wave of amplitude Eo, frequency and
the phase angle zero that is travelling in the direction from the origin to
‘the point (1, 1, 1) with polarization parallel to the xz-plane.
Show that the electromagnetic potentials in uniform electric and
magnetic fields may be expressed as
> >
o=E.r andA =5B xT),
where r. is the position vector of the point under consideration.
Using the substitution z = ei9, where @ is real and positive, and the
residue theorem, evaluate the integral
2n
ea
) (2+sin0)
A spherical shell of radius R, carrying a uniform surface charge
density o, is set spinning at angular velocity o. Calculate its magnetic
dipole moment. Show that, for r >> R, the magnetic vector potential of
the shell is that of a perfect dipole.
The radiation resistance of an antenna is 80 Q and loss resistance is
10 Q. Find its directivity if power gain is 20.
ay
Using the laws of transformation of the electric field E and the
>
magnetic field B , under Lorentz transformations, show that the
-> >
combination (E 2_ 238 ‘ is relativistically invariant.
In a Young’s double-slit experiment, the separation between the
second-order bright fringe and the central bright fringe, Y, on a flat
screen is 0:0180 m, when the light has a wavelength of 425 nm. Assume
that the angles, 0, that locate the fringes on the screen are so small that
sin @ ~ tan 0. Calculate the separation Y when the wavelength of light is
585 nm.
What is GPS augmentation ? Why is it required ? Name a few of the
GPS augmentation systems.
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Q10. (a)
(b)
(c)
SGSE-P-GPH
What would be the height of the atmosphere if the air density
(63) were uniform,
(ii) decreased linearly to zero with height ?
Assume that at sea level the air pressure is 2-0 atm and the air density
is 1:3 kg/m?. [Given g = 9-8 m/s?|
Find the Laplace transform of the function
ett
-—
F(t) =
For some spherically symmetric distribution of charges, the potential on
the surface of radius R is given to be Vo- Calculate the potential in the
region r > R. Also calculate the potential for r < R, when no charges are
present inside the sphere.
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