8/13/2019 EM List of Problems

http://slidepdf.com/reader/full/em-list-of-problems 1/1

Relativity

1.  Uniform Ek in the x-direction and an induced Bk = 2Ek . The induced Bk lies in the x-y plane

at an angle alpha with the x axis. Find the velocity such that in this frame of reference Bk 

and Ek are parallel. Also, in this new frame, What are the fields’ magnitudes and directions

if (a) alpha << 1 (b) alpha -> pi/2.2.  A straight cable infinite in length and of negligible thickness is at rest and has linear charge

density q in the intertial frame K’. In the frame K the calbe is moving with velocity v 

parallel to its axis. (a) Write the E and B field in cylindrical coordinates in the rest frame K’

and then use Lorentz transformations to find the fields in K. (b) What are the charge

density and current you found in part (a). Calculate the electric and magnetic fields

directly in K’ and compare calculations 

3.  Write Maxwells eqns in terms of the 4-potential in he Lorentz guage. (a) supposing there

are no sources, solve these differential eqns supposing Au

= Au(x,t) . (b) Sove the equations

with sources using retarded potentials.

4.  Given the 4-potential Au

= Au

(ct3

,xt,-yt, -3zt2

) …. (a) calculate E and B (b) Do a“tranformacion de norma” using the función f= xyt. Find the new 4-potential. (c) How did

E and B change.

5.  E and B in one frame form an angle of 60 degrees. The E field has magnitude ‘a’ and the B

field has magnitude of 3a.. In another frame, B’ has a magnitude of 4a. Find the

magnitude of E’ and the angle they form in this frame. 

6.  Construct the possible invariants of Fuv

. ( See Griffiths Ch.12)

a.  True or False: If E and B are orthogonal in one frame then they both must be

nonzero in all frames.

b.  True or False. A purely B field can change to purely E and vice versa in another

frame.c.  True or False. If E and B are orthogonal, and B>E we can get E’ = 0 

d.  True or False. If E and B are orthogonal and E>B, we can get B’ = 0 

7.  Show that the wave equation is not invariant under Galilean Transformation but it is

invariant under Lorentz Transformation.

8.  A charged particles is radiating. Deduce the relativistic expression for the power radiated.

Hint: use Larmors formula and substitute the momentum. Based on what you obtained.

9.  Imagine we have a loop of wire without charge in the shape of a rectangle of sides lengths

a and b. The loop carries a current I0 and the whole loop moves with uniform velocity V 

parallel to side a and the x axis. The wire has a finite cross section. Find the distribution of

electric charges on the loop and their magnetic moments in the lab system frame.