#### 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

\begin{equation} \frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right) = \frac{\partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B} + {\boldsymbol E} \times \frac{\partial {\boldsymbol B}}{\partial t}. \end{equation}

Using Faraday to substitute for \(\frac{\partial {\boldsymbol B}}{\partial t}\),

\begin{equation} \frac{ \partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B} = \frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B}\right) + {\boldsymbol E} \times \left({\boldsymbol \nabla} \times {\boldsymbol E} \right) \end{equation}

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 grad_sprod, \[ \frac{1}{2}{\boldsymbol \nabla} \left( E^2 \right) = \left( {\boldsymbol E} \cdot {\boldsymbol \nabla} \right) {\boldsymbol E} + {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right) \] 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

Maxwell stress tensor

\begin{equation} T_{ij} \equiv \varepsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right) \tag{MaxST}\label{MaxST} \end{equation}

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

We then obtain the

EM force per unit volume

\[ {\boldsymbol f} = {\boldsymbol \nabla} \cdot {\boldsymbol T} - \varepsilon_0 \mu_0 \frac{\partial {\boldsymbol S}}{\partial t} \tag{fT}\label{fT} \]

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

Total force on charges in volume

\[ {\boldsymbol F} = \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} - \varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau. \tag{totFo}\label{totFo} \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42