# Field Theory January 2008

Note : 1. Answer any FIVE full questions.

2. Assume any missing data suitably.

1 a. State and explain Gauss Law. Find out the relation between D and E.

b. Charge is distributed uniformly along an infinite straight line with constant density .Develop the expression for E at the general point P.

c. A vector field is given by, A {r,§,z) = 2>Qe~’ a -2za_. Verify the divergence theorem for the volume enclosed by, r = 2, z – 5 .

2 a.If E = -8xya^ — 4x a ^ + a (V/m). Find the work done in carrying a 6 coulomb charge from A(l, 8, 5) to B(2, 18,,6) along the path y = 3x + 2, z = x + 4.

b. A potential function is v=2x+4y volts, is in free space. Find the stored energy in free space in the 1 m3 volume centered at origin.

c. Starting with principle of charge conservation, obtain point form of continuity equation.

3 a. Obtain the conditions on the tangential and normal components of electric field intensity and electric flux density at the boundary between two dielectric media.

b. Derive Poisson’s and Laplace’s equations starting from point form of Gauss law.

c. State and explain uniqueness theorem.

4 a. Find H at the centre of a square current loop of side 4 meters, if a current of 5 amp is passing through it.

b. State and explain Ampere’s circuit law.

c. Given A = (ycosax)ax +{y+ex)a _, find VxA atthe origin.

5 a. Derive Lorentz force equation and mention the application of its solution.

b. Define torque. Find the torque about the z-axis for a conductor located at x = 0.4 m, y = 0 and 0 < z < 2m, which carries a current of 5A in the a_ direction, along the length of the conductor. B = 2.5a Tesla.

C . Derive the expression for the inductance of a toroidial ring with N turns and carrying current 1 amp. Assume the radius of the toroid be ‘R’m and area of cross section of toroidial ring be ‘A’m2.

6 a. State and explain Faraday’s law for EMF when a closed conductor single loop circuit is placed in time varying magnetic field and hence show that V x E

b. Write Maxwell’s equations for free space in point and integral forms.

c. Write a short note on retarded potentials.

7 a. What is uniform plane wave? Explain its propagation in free space with necessary equations.

b. What is loss tangent? Explain its practical importance.

C. Find the skin depth 8 at a frequency of 1.6 MHz in aluminium, where a = 38.2 MS/m and jo, = 1. Also find y, X and Vp .

8 a. Define the terms

i)   Reflection co-efficient and

ii) Transmission co-efficient. Also bring out the relation between them.

b. Write a short note on SWR.

c. A 50 MHz uniform plane wave has electric field amplitude 10 V/m. The medium is lossless, having e^ = 9 and \i r= 1. The wave propagates in the x, y plane at a 30° angle to the x axis and is linearly polarized along z. Write down the phasor expression for the electric field. Also find Xx, Xy, \px and Npy.