91Ó°ÊÓ

At a given instant, a certain system has a current density given by \(\mathbf{J}=A\left(x^{3} \hat{\mathbf{x}}+y^{3} \hat{\mathbf{y}}+z^{3} \hat{\mathbf{z}}\right)\) where \(A\) is a positive constant. (a) In what units will \(A\) be measured? (b) At this instant, what is the rate of change of the charge density at the point \((2,-1,4)\) meter? (c) Consider the total charge \(Q\) contained within a sphere of radius \(a\) centered at the origin. At this instant, what is the rate at which \(Q\) is changing in time? Is \(Q\) increasing or decreasing?

A total charge \(Q\) is distributed uniformly throughout a sphere of radius \(a\). The sphere is then rotated with constant angular speed \(\omega\) about a diameter. Assume that the charge distribution is unaffected by the rotation, and find \(\mathbf{J}\) everywhere within the sphere. (Express it in spherical coordinates with the polar axis coinciding with the axis of rotation.) Find the total current passing through a semicircle of radius \(a\) fixed in space with its base on the axis of rotation.

Two large parallel plane conducting plates are a distance \(d\) apart. The region between them is filled with two l.i.h. materials whose surface of separation is a plane parallel to the plates. The first material (with properties \(\sigma_{1}\) and \(\epsilon_{1}\) ) is of thickness \(x\), while the second material \(\left(\sigma_{2}, \epsilon_{2}\right)\) has thickness \(d-x\). There is a steady current between the plates that are kept at constant potentials of \(\phi_{1}\) and \(\phi_{2}\). Find the potential at the surface of separation of the two media, and the free surface charge density there.

Show by direct integration that when a capacitor is charged by a battery of emf \(\mathscr{E}\), the amount of heat developed in the circuit is equal to the final electrostatic energy of the capacitor.