# Pre-Quantum Electrodynamics
Author: Jean-Sébastien Caux

#### Maxwell's Stress Tensor
emd.ce.mst

Total EM force on charges in volume \({\cal V}\): \[ {\boldsymbol F} = \int_{\cal V} \left( {\boldsymbol E} + {\boldsymbol v} \times {\boldsymbol B} \right) \rho ~d\tau = \int_{\cal V} \left( \rho {\boldsymbol E} + {\boldsymbol J} \times {\boldsymbol B} \right) d\tau \] Force per unit volume: \[ {\boldsymbol f} \equiv \rho {\boldsymbol E} + {\boldsymbol J} \times {\boldsymbol B}. \] Substitute for \(\rho\) and \({\boldsymbol J}\) using Maxwell (Gauss and Ampère-Maxwell): \[ {\boldsymbol f} = \varepsilon_0 ({\boldsymbol \nabla} \cdot {\boldsymbol E}) {\boldsymbol E} + \left( \frac{1}{\mu_0} {\boldsymbol \nabla} \times {\boldsymbol B} - \varepsilon_0 \frac{ \partial {\boldsymbol E}}{\partial t} \right) \times {\boldsymbol B}. \]

On the other hand we have \[ \frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B} \right) = \frac{∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B}

- {\boldsymbol E} × \frac{∂ {\boldsymbol B}}{∂ t}.

\] Using Faraday to substitute for \(\frac{\partial {\boldsymbol B}}{\partial t}\), \[ \frac{ ∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B} = \frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B}\right)

- {\boldsymbol E} × \left({\boldsymbol ∇} × {\boldsymbol E} \right)

\] so \[ {\boldsymbol f} = \varepsilon_0 \left( \left( {\boldsymbol \nabla} \cdot {\boldsymbol E} \right) {\boldsymbol E} - {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right) \right) - \frac{1}{\mu_0} \left( {\boldsymbol B} \times \left( {\boldsymbol \nabla} \times {\boldsymbol B} \right) \right) - \varepsilon_0 \frac{\partial}{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right). \] Since \({\boldsymbol \nabla} \cdot {\boldsymbol B} = 0\), we can symmetrize the expression in \({\boldsymbol E}\) and \({\boldsymbol B}\). Moreover, by product rule 4, \[ \frac{1}{2}{\boldsymbol \nabla} \left( E^2 \right) = \left( {\boldsymbol E} \cdot {\boldsymbol \nabla} \right) {\boldsymbol E} + {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right) \] so \[ {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right) = \frac{1}{2} {\boldsymbol \nabla} E^2 - \left({\boldsymbol E} \cdot {\boldsymbol \nabla} \right) {\boldsymbol E} \] and similarly for \({\boldsymbol B}\). We thus get

\begin{align} {\boldsymbol f} =& \varepsilon_0 \left( \left( {\boldsymbol \nabla} \cdot {\boldsymbol E} \right) {\boldsymbol E} + \left( {\boldsymbol E} \cdot {\boldsymbol \nabla} \right) {\boldsymbol E} \right) + \frac{1}{\mu_0} \left( \left( {\boldsymbol \nabla} \cdot {\boldsymbol B} \right) {\boldsymbol B} + \left( {\boldsymbol B} \cdot {\boldsymbol \nabla} \right) {\boldsymbol B} \right) \\ &- \frac{1}{2} {\boldsymbol \nabla} \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) - \varepsilon_0 \frac{\partial}{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right). \end{align}

This expression can be greatly simplified by introducing the

{\bf Maxwell stress tensor} \[ T_{ij} ≡ ε_0 \left( E_i E_j - \frac{1}{2} δ_{ij} E^2\right)

- \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} δ_{ij} B^2 \right)

\]

The element \(T_{ij}\) represents the force per unit area in the $i$th direction acting on a surface element oriented in the $j$th direction. Diagonal elements are pressures, off-diagonal elements are shears.

We then obtain

{\bf EM force per unit volume} \[ {\boldsymbol f} = {\boldsymbol \nabla} \cdot {\boldsymbol T} - \varepsilon_0 \mu_0 \frac{\partial {\boldsymbol S}}{\partial t} \]

where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the

{\bf Total force on charges in volume} \[ {\boldsymbol F} = \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} - \varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau. \]

Author: Jean-Sébastien Caux

Created: 2022-02-08 Tue 06:55

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