Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 3 / 17A (Electromagnetism)

Choose $+z$ along the current in the inner conductor. Cylindrical symmetry and Ampère's law give $2\pi r{B}_{\varphi }={\mu }_{0}I$ for $a<r<b$, soThe opposite current on the outer shell makes the net enclosed current zero outside both conductors; the long-cylinder approximation neglects end fields.

Take the inner conductor to be at potential $V$ above the outer. A cylindrical Gauss law surface of length $\ell$ encloses charge $\lambda \ell$ and gives $E_{r}2\pi r\ell =\lambda \ell /{ϵ}_0$. The potential difference isThus the coaxial cylindrical capacitor field and charge areThe perfectly conducting surfaces have no tangential electric field in this idealized steady model.

The Poynting vector in the annular region isIt points from the battery towards the resistor with the preceding current convention. Integrating through an annular cross-section givesThe energy is transported by the electromagnetic field in the space between the ideal conductors; the integration recovers the power dissipated at the terminating resistor.