1. An long straight wire  lies along the z direction. At time , a constant current  is switched on.  Calculate the vector potential and the electric and the magnetic fields.

The problem is very similar to Problem 2 of the Tutorial. The vector  potential is given by

2. A conducting wire is bent  into a loop of two semicircles, one of radius  and the other connected by straight segments. The loop carries a current  , which is directed anticlockwise on the shorter semicircle and clockwise on the other one.. Calculate the vector potential at the centre of the loop. Calculate the electric field at the centre. Can you calculate the magnetic field from the expression of the vector potential?

 

 

          

Since the current is switched on at t=0, the (retarded) vector potential at the centre O is given by



Since , the second integral vanishes. We are left with . Note that the integral is a line integral over the loop, which is directed differently at different points of the loop. Let us take the loop to be in the x-y plane with directions as shown in the figure. Recall that the direction of   is along the direction of the current. The contribution from the two straight sections give , the negative sign  is due to the fact that the current in both the sections are directed along the negative x direction.
We now have to perform the integration over the two semicircles.   Consider the integration over the bigger semicircle.  We can resolve . The denominator of the integral also has  so that the integral becomes independent of  (In a similar way, the   integral of the smaller semicircle will be independent of ).  The y-component of the integral would vanish by symmetrically placed elements on either side of the y-axis, leaving us with   as the value of the integral. The integral over the smaller semicircle is done in a similar way but its direction would be opposite because of the direction of the current. Thus the contribution from the two semicircles would cancel and the vector potential at the centre would be given by  . The electric field is trivial and is given by


Since the vector potential has been calculated only at a single point, it is not possible to calculate its curl and hence the magnetic field cannot be determined by the above calculation.