Introduction to Electrodynamics/Chapter 7/5/3

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7.5.3 Maxwell's Stress Tensor

The total force acting on charges inside a volume V:

<math>\vec{F} = \int_V (\vec{E} + \vec{v} \times \vec{B})\rho\ d\tau = \int_V (\rho\vec{E} + \vec{J}\times\vec{B})\ d\tau\;</math>

Force per unit volume:

<math>\vec{\mathcal{F}} = \rho\vec{E} + \vec{J}\times\vec{B}\;</math>

Let's express this in terms of E and B alone.

Gauss's Law:

<math>\rho = \epsilon_0 \nabla \cdot \vec{E}\;</math>

Ampere's Law with Maxwell's Correction:

<math>\vec{J} = {1 \over \mu_0} \nabla \times \vec{B} - \epsilon_0 {\partial \vec{E} \over \partial t}\;</math>

Plug it in:

<math>\vec{\mathcal{F}} = \epsilon_0 (\nabla \cdot \vec{E}) \vec{E} + \left( {1 \over \mu_0} \nabla \times \vec{B} - \epsilon_0 {\partial \vec{E} \over \partial t} \right) \times \vec{B}\;</math>
<math>\vec{\mathcal{F}} = \epsilon_0 (\nabla \cdot \vec{E}) \vec{E} + {1 \over \mu_0} (\nabla \times \vec{B}) \times \vec{B} - \epsilon_0 {\partial \vec{E} \over \partial t} \times \vec{B}\;</math>

Notice:

<math>{\partial \over \partial t} (\vec{E} \times \vec{B}) = {\partial \vec{E} \over \partial t} \times \vec{B} + \vec{E} \times {\partial \vec{B} \over \partial t}\;</math>

Faraday's Law:

<math>{\partial \over \partial t} (\vec{E} \times \vec{B}) = {\partial \vec{E} \over \partial t} \times \vec{B} - \vec{E} \times (\nabla \times \vec{E})\;</math>

Reorder terms:

<math>{\partial \vec{E} \over \partial t} \times \vec{B} = {\partial \over \partial t} (\vec{E} \times \vec{B}) + \vec{E} \times (\nabla \times \vec{E})\;</math>

Plug it in to our force per unit volume equation:

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