Electromagnetic Fields and Transmission Lines

An electric potential is given by, V = 60 sin /r² volt. Find V and E at P[3, 60°, 25^o).

Write & explain:

(i) Faraday's law.

Derive the condition for a distortion less line and comment on the result. 

Given the flux density D = (2 cos/r³) a_r + (sin /r³)  C/m², evaluate both sides of the divergence theorem for the region defined by . 

Find the volume charge density that is associated with

Let V = 2xy²z³+ 3 In (x² + 2y² + 3z²) V in free space. Evaluate each of the following quantities at P[3, 2, -1):

(1) V    (ii)  (iii) E.

Assuming the potential function V varies as a function of  in cylindrical coordinates systems, obtain the solution of Laplace equation and deduce the value of capacitance of a coaxial capacitor.

A charge of 10 nC is moving with a velocity of 10⁷ (-0.5 a_x + a_y -0.71a_z) m/s. Determine the force exerted on the test charge when

(i) a magnetic induction B = (a_x + 2 a_y + 3 a_z) Wb/m² is applied

(ii) an electric field E = (3a_x + 2a_y + a_z) kV/m is applied.

(iii) When B and Egiven above are applied simultaneously. 

Using Faraday's law and the concept of displacement current density,Ampere's circuital law, divergence theorem, obtain all Maxwell's equations for time varying fields in point and integral forms. Derive the necessary equations. 

The region y < 0 contains a dielectric material for which =2.5,while the region y >O is characterized by = 4. Let E1 =30 a_x + 50 a_y + 70 a_z V/m Find (1) DN₂ (ü) D_t2 (iii) D₂ (iv) . 

Explain boundary condition for dielectric material.

Write & explain:

(ii) Ampere's circuital law.

State and prove Poynting theorem relating to the flow of energy at a point in space in an electromagnetic field. 

A wave propagating in a lossless dielectric has the components E= 500 cos   V/m and H = 1. 1 cos   A/m. If the wave is travelling with the velocity of U = 0.5c, find (i) Lr (ii)  (iii)  (iv)  (v) 

Explain Skin effect in detail. A steel pipe is constructed of a material for which  = 200 and  = 5 x 106 mho/m. The outer and inner radii 8 and 6 mm respectively and the length is 80 m. If the total current carried by the pipe is 2 cos 10⁴  A, find (i) the skin depth (ii) the effective resistance. 

Prove for a travelling uniform plane waves  = 120 ohm.where E & H are amplitudes of electric and magnetic field.

What is stub matching? Outline the solution for the single stub matching problem. 

Draw the equivalent diagram of transmission line of length L. hence obtain the secondary constant, primary constant and their relationship between them

A 50 m long loss less transmission line with characteristic impedance of 50 ohm operating at 2 MHz is terminated with a load 30 - j23 , find:

(i) The reflection coefficient, (K).

(ii) The voltage standing wave ratio, (S). 

Derive expression for attenuation constant  and phase constant  for lossless transmission lines.

Derive continuity equation of current also explain relaxation time.

Write Laplace equation in spherical coordinate and cylindrical coordinate system.