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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 15D / a / Solution

In a region with no current, the Ampère-Maxwell equation and Faraday's law are

\nabla \times B = μ_{0} ϵ_{0} \frac{\partial E}{\partial t}, \nabla \times E = - \frac{\partial B}{\partial t} .

Therefore

\frac{\partial w}{\partial t} = ϵ_{0} E \cdot \partial_{t} E + \frac{1}{μ _{0}} B \cdot \partial_{t} B = \frac{1}{μ _{0}} [E \cdot (\nabla \times B) - B \cdot (\nabla \times E)] = - \frac{1}{μ _{0}} \nabla \cdot (E \times B),

where the last line uses the divergence and curl of a cross product. Thus Poynting's theorem takes the form

\frac{\partial w}{\partial t} + \nabla \cdot S = 0, S = \frac{1}{μ _{0}} E \times B .

The vector

S

is the Poynting vector.

Solved by gpt-5.6-sol high.

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