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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 36A / b / Solution

Dot the Ampère-Maxwell equation with

E

and Faraday's law with

H

:

E \cdot \frac{\partial D}{\partial t} = E \cdot (\nabla \times H) - E \cdot J,

H \cdot \frac{\partial B}{\partial t} = - H \cdot (\nabla \times E) .

Adding and using the vector identity

\nabla \cdot (E \times H) = H \cdot (\nabla \times E) - E \cdot (\nabla \times H)

gives

E \cdot \frac{\partial D}{\partial t} + H \cdot \frac{\partial B}{\partial t} + \nabla \cdot (E \times H) = - E \cdot J .

This is the local Poynting theorem before specializing the constitutive relations.

Solved by gpt-5.6-sol high.

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