1. Write down the elements of the  stress tensor for an infinite solenoid carrying current I. What is the pressure on the curved surface of the solenoid whose cross section has a radius R?

In a solenoid there is no electric field and the magnetic field is in the axial direction, taken as the z direction. Let the field inside be . Since all the off diagonal components are zero. We use,

so that , and .The Stress Tensor matrix is given by

The force on an area is . The area element on the curved surface is directed along the inward radial direction and is given by . Since the stress tensor is written in terms of Cartesian components,  we express  the cylindrical radial vector of a polar system as , so that .

Dividing by the area element , the pressure is outward and is given by .

2. Two opposite charges are at a distance q apart. Calculate the force between them using the stress tensor. (This is of course trivial but is primarily to test understanding of the method).

We will be calculating the force by integrating the stress tensor. In order to do so, we need the electric field expressions due to the two charges and also need to define a surface. Let us enclose the positive charge by a disk  in the median plane and a hemisphere  over it. The radius of the disk and hemisphere will be taken to be infinitely large so that the fields vanish. As a result the integration over the curved surface gives zero.

The net electric field on the median plane must be towards left  (along .Thus only needs to be considered.

If we take  an annular region at a distance y from the midpoint, the field  strength  is

Thus and the surface integral is