Question

Given the Minkowski momentum density gy and the Maxwell's stress tensor T,whose components are given by T0 = E̟1 D3, + H1B3 - 1/2 (ED+HB) s4 derive the \text { hydrodynamical equation of motion } \frac{\partial \mathbf{g}_{M}}{\partial t}=\nabla \cdot \mathbf{T}-\mathbf{F}, \text { where } \mathbf{F}=\rho \mathbf{E}+\mathbf{J} \times \mathbf{B} \text { is the } Lorentz force exerted on the distribution of charges and currents by the electromagnetic field. You may assume no spatial dependence to u and ɛ. NB. These are the full spatiotemporal fields denoted by script notation in the notes. HINT: 1/2V(E-E)=Ex(V×E)+(E-V)E.

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