Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 15D / c

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- Solution c

Solution

Dot the Ampère-Maxwell equation with

E

, dot Faraday's law with

B / μ_{0}

, and use the stated vector identity. The result is the local balance law

\frac{\partial}{\partial t} [\frac{1}{2} (ϵ_{0} E^{2} + \frac{B ^{2}}{μ _{0}})] + \nabla \cdot S = - J \cdot E, S = \frac{1}{μ _{0}} E \times B .

Here

S

is the Poynting vector. Integration over

V

gives the Poynting theorem

\frac{d U}{d t} = - \int_{\partial V} S \cdot n d S - \int_{V} J \cdot E d^{3} x .

The field energy decreases through outward electromagnetic energy flux and through work done on charges.

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