Question

An uniform magnetic field B is along z-axis. Find the corresponding vector potential A which has

(a) only the x-component.

(b) only the y-component.

(c) both x- and y-components and write your result in terms of r × B.

(d) Do your results in (a), (b) and (c) satisfy Coulomb/London gauge ∇ · A=0?

The gauge (a) or (b) is called the Landau gauge and the gauge in (c) is called symmetric gauge.

Answer 1

And the vector potential is
  
So, the z-component is

a).
  So, the vector potential which has only x component is
  
b)
   So, the vector potential which has only y component is
  
c)
   So, the vector potential which has x and y both components is
  
And when written in terms of cross product, we write
   
d)
As
    for all three casses, so, all three results satisfy Coulomb/ London gauge.