Paper 3, Section II, C

Vector Calculus | Part IA, 2013

If E\mathbf{E} and B\mathbf{B} are vectors in R3\mathbb{R}^{3}, show that

Tij=EiEj+BiBj12δij(EkEk+BkBk)T_{i j}=E_{i} E_{j}+B_{i} B_{j}-\frac{1}{2} \delta_{i j}\left(E_{k} E_{k}+B_{k} B_{k}\right)

is a second rank tensor.

Now assume that E(x,t)\mathbf{E}(\mathbf{x}, t) and B(x,t)\mathbf{B}(\mathbf{x}, t) obey Maxwell's equations, which in suitable units read

E=ρB=0×E=Bt×B=J+Et\begin{aligned} &\nabla \cdot \mathbf{E}=\rho \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t} \end{aligned}

where ρ\rho is the charge density and J\mathbf{J} the current density. Show that

t(E×B)=MρEJ×B where Mi=Tijxj\frac{\partial}{\partial t}(\mathbf{E} \times \mathbf{B})=\mathbf{M}-\rho \mathbf{E}-\mathbf{J} \times \mathbf{B} \quad \text { where } \quad M_{i}=\frac{\partial T_{i j}}{\partial x_{j}}

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