ELECTRIC FIELDS IN MATTER — Edward Purcell & David Morin
Physics — E&M · em-waves-poynting-vector
Problem
A very thin straight wire carries a constant current I from infinity radially inward to a spherical conducting shell with radius R. The increase in the charge on the shell causes the electric field in the surrounding space to increase, which means that the energy density increases. This implies that there must be a flow of energy from somewhere. This ``somewhere'' is the wire. Verify that the total flux of the Poynting vector away from a thin tube surrounding the wire equals the rate of change of the energy stored in the electric field. (You can assume that the radius of the wire is much smaller than the radius of the tube, which in turn is much smaller than the radius of the shell.)