Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 12B / a

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- Solution a

Solution

The vector identity from Question 3 gives

\nabla \cdot (E \times B) = B \cdot (\nabla \times E) - E \cdot (\nabla \times B) = - B \cdot \frac{\partial B}{\partial t} - E \cdot J - E \cdot \frac{\partial E}{\partial t} = - E \cdot J - \frac{\partial}{\partial t} \frac{∣ E ∣ ^{2} + ∣ B ∣ ^{2}}{2} .

Integrating over

V

and applying the divergence theorem proves the Poynting theorem

\int_{S} P \cdot d S = - \int_{V} E \cdot J d V - \frac{\partial}{\partial t} \int_{V} \frac{∣ E ∣ ^{2} + ∣ B ∣ ^{2}}{2} d V .

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