# Field Theory December 2010

Note: 1. Answer any FIVE full questions, selecting at least TWO questions from each part.

2. Assume any missing data suitably.

PART-A

1 a. Show that the electric field intensity at a point, due to sn5 number of point charges, is given – 1 AO.

b.  A uniform line charge of infinite length with pL = 40 nc/m , lies along the z-axis. Find E at (-2, .2, 8) in air.

c.   State and prove the Gauss’s law.

d.  Determine the volume charge density, if the field is D=ar c/m .

2 a. Derive an equation for the potential at a point, due to an infinite line charge.

b.  If the potential field V = 3x2 + 3y2 + 2z3 volts, find

i) V ii) E iii) D at P(-4,5,4)

c.   Deduce an equation for the capacitance of a coaxial cable of length *L’, radius of inner conductor ‘a’ and out conductor 4b.

3 a.  State and prove the uniqueness theorem.

b. Find the capacitance between the two concentric spheres of radii r = b and r = a, such that b > a, if the potential V – 0 at r – b, using the Laplace’s equation.

c.   Determine whether or not the potential equations i) V – 2x2 ~ Ay1 + z2 and ii) V = r2cos<|> + 0 satisfy the Laplace’s equation.

4 a.  State and prove the Stoke’s theorem.

b.  If the magnetic field intensity in a region is H = (3y – 2)az + 2xay, find the current density at the origin.

c.   A co-axial cable with radius of inner conductor a, inner radius of outer conductor b and outer radius c carries a current I at inner conductor and -I in the outer conductor. Determine and sketch variation of H against r for i) r < a ii) a < r < b iii) b < r < c iv) r > c.

PART-B

5    a. Derive an equation for the force between the two differential current elements.

b.  Derive the magnetic boundary conditions at the interface between the two different magnetic materials. Discuss the conditions.

c.   Calculate the inductance of a solenoid of 400 turns wound on a cylindrical tube of 10 cm diameter and 50 cm length. Assume the solenoid is in air.

6    a. Using the Faraday’s law, deduce the Maxwell’s equation, to relate time varying electric and magnetic fields.

b.   Derive the Maxwell ’ s equations in the point form of the Gaus s ’ s law for time varying fields.

c.   Given E = Em sin(cot – pz)a in free space. Find D , B and H.

7    a. Obtain the solution of wave equation for uniform plane wave in free space.

b. State and explain the Poynting’s theorem.

c.  For a wave traveling in air, the electric field is given by E = 6cos(cot – (3t)az at f = 10 MHz. Calculate the average Poynting vector.

8    a. Explain the reflection of uniform plane waves, with normal incidence at a plane dielectric boundary.

b.  Write short notes on:

i) Standing wave ratio.

ii) Skin effect in conductors.