1. A coaxial cable consists of an inner conductor of radius a and an outer conductor of radius b. The current in the inner conductor is uniformly distributed while the outer conductor has negligible thickness and provides a return path to the current. Calculate the self inductance of the cable.

The self inductance of the inner conductor was calculated in Problem 1. Because the thickness of the outer conductor is negligible, the flux linked is zero and there is no self inductance contribution from the outer shell. There is however, flux linked between the two conductors. The field in this region is , so that flux passing through a strip of width dr between r and r+dr and of unit length is . The total flux linked is . Thus the total self inductance is

2. Find the mutual inductance of two coplanar squares with a common centre  assuming that the square located inside is much smaller in dimension b than the bigger square which has a side a.

The magnetic field at the centre of the square (taken in xy) plane) is given by . Since the dimension of the second coil is much smaller than that of the first, it can be assumed that this field exists over the entire square of side b,  as a result of which the flux through the second loop is , which gives the mutual inductance to be  .

3. The current through a straight wire varies with time as  where the current is in Amperes and time in seconds. If the radius of cross section of the wire is 50 (mm)² and the resistivity  , estimate the displacement current density and compare it with the conduction current density.

Consider a length L of the wire. The resistance is . If E is the electric field in the wire, the potential difference between the ends of the wire is  so that the current . The displacement current density is

Thus one sees that the displacement current density is very small, the corresponding current   density at t=0 is  A/m2 which keeps on increasing with time.

4. A capacitor plate of area 0.3 m² is being charged at a uniform rate so that the electric field inside the plate varies with time as  V/m-s. Calculate the displacement current and estimate the magnetic field strength at a distance 5 cm from the centre of the capacitor plate along a line parallel to both the plates.

The displacement current  density 

The magnetic field at a distance r from the centre is given by

which gives, substituting r= 0.5 m,