Electromagnetic Field and Theory 2011-12

Note (t) All questions carry equal marks.

(2) Attempt all questions.

1. Attempt any four parts of the following :

(a) Given vector field G = 8 sin (p ar in spherical coordinate Transform it into:

(i) Rectangular Coordinate

(ii) Cylindrical Coordinate

(b) Find the gradient of the following scalar field

(i) V = e~z sin 2x cosh y

(ii) U = P2 Z cos 2tp

(c) Derive expression for electric field due to infinitely long wire.

(d) Write down the expression/statement of the :

(i) Maxwell’s curl equation for time varying field.

(ii) Differential form of Ampere’s law.

A = (3x+4z) ax + (c2x-5z) ay + (4x-c,y+c4z) az Calculate c,, c2, c3, c4 if A is irrotational and solenoidal.

(f)  A charge distribution with spherical symmetiy has density pv= p0 r/R, 0 < r < R and 0 for r > R, Determine E everywhere.

2.  Attempt any two parts of the following :

(a) Derive Energy’ density in electrostatic field. A sphere of volume 0.1 m3 has a charge density of 8.0pc /m3. Find the electric field at a point (2,0,0) if the centre of the sphere is at (0,0,0).

(b) State and explain the Coulomb’s law. If the current density J=l/r2 (cos 0a+ sin 9 a0). A/m2, find the current passing through a sphere of radius 1.0 m.

(c) Discuss the relevance of uniqueness theorem. A spherical condenser has capacity of 54 pF. It consist of two concentric sphere differing in radius by 4 cm and having an air as dielectric. Find their radii.

3. Attempt any four parts of the following :

(a) Write down the boundary condition for current density and postulates of Magnetostatics in free space.

(b) Prove that B = (|^0 Ife2/4R3) (aR2 cos 0 +a0 sin 0) for magnetic dipole.

(c)  Given that -2ax+6av+4a? A/m in region y-x-2<0 where M-,=5|^0 calculate M, and Br

(d) Find inductance of coaxial cable.

(e) Explain the relevance of Magnetic scalar and vector potential.

(f)   In a material for which a=5 S/m and er=l, the electrical field intensity is E=250 sin 1010t V/m. Find conduction and displacement current densities and the frequency at which both have equal magnitude.

4. Attempt any two parts of the following :

(a) Derive the expression for a and P in a conducting medium. Explain skin effect and depth of penetration.

(b) Derive the wave equation for conducting media. A uniform plane wave is propagating in the +z direction in a good conductor having conductivity cr S/m. The permittivity and permeabi lity In the conductor are the same as in free space and the electric field is xE0 at z = 0. What power (W/nr) is C dissipated in this medium foe z > 0 ? Assume cr »cos.

(c) Derive Faraday law of induction. Explain the concept of Transformer and motional electromotive force. Discuss the relevance of Anisotropic media.

5. Attempt any two parts of the following :

(a) Discuss the structure of Smith Chart. How it is used for measurement of impedances and VSWR ?

(b) Relate short circuit, open circuit and characteristic impedance of Transmission line. The short circuit and open circuit impedance of 10 km long open wire transmission line are Z =2930 Z26° and Z = 260 Z-32° at a frequency

of 1 kHz. Calculate the characteristics impedance and phase velocity.

(c) Define reflection loss, transmission loss and return loss. The 600 Q lossless transmission line is fed by 50 D, generator. If the line is 200 meter long and terminated by load 500 Q. Determine in db

(i) Reflection loss

(ii) Transmission loss

(iii) Return loss.