Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 6 / b / i

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- Solution i

Solution

Apply the symmetric gradient and curl decomposition separately to the electric field and the magnetic field:

S_{ij}^{E} = \frac{1}{2} (\partial_{i} E_{j} + \partial_{j} E_{i}), S_{ij}^{B} = \frac{1}{2} (\partial_{i} B_{j} + \partial_{j} B_{i}) .

The corresponding axial parts are

u^{E} = \frac{1}{2} \nabla \times E = - \frac{1}{2} \partial_{t} B, u^{B} = \frac{1}{2} \nabla \times B = \frac{1}{2 c ^{2}} \partial_{t} E .

The second equalities use the vacuum Maxwell equations. Their divergence constraints also show that both symmetric gradient tensors are traceless.

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