Electrodynamics/Electrostatic Stress Tensor

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Force on a Charge[edit]

When we want to discuss the force on a charge due to a charge distribution, there are two options. The first is a more traditional method: an integral over a volume containing the charge distribution. The second method is less traditional but is easier to do: a surface integral over a special stress tensor.

Volume Integral Version[edit]

\mathbf{F} = \int_V \rho \mathbf{E} dV

Electrostatic Stress Tensor[edit]

\mathbb{T}_E = \frac{1}{4 \pi} \begin{bmatrix}
 E_x^2 - \frac{E^2}{2} & E_x E_y & E_x E_z \\
 E_x E_y & E_y^2 - \frac{E^2}{2} & E_y E_z \\
 E_x E_z & E_y E_z & E_z^2 - \frac{E^2}{2}
\end{bmatrix}

Surface Integral Version[edit]

\mathbf{F} = \int_S \mathbb{T} \mathbf{n} dA

The Maxwell Stress Tensor[edit]

Tij is called the Maxwell Stress Tensor, it has two indices and is not a vector so is given a double arrow.

\mathbb{T}_{ij} = \epsilon_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)