❓What is the Maxwell stress tensor and how is it used to calculate the force on charges?

The Maxwell stress tensor ($\mathbf{\sigma}$ or $\mathbf{T}$) is a symmetric second-order tensor used in electromagnetism to calculate the force exerted by the electromagnetic field on charges within a volume. In the static (electrostatic) case with no magnetic field ($B=0$), the stress tensor components used for the surface force calculation are defined as: $\sigma_{i3} = \epsilon_0 \left(E_i E_3 - \frac{1}{2} E^2 \delta_{i3}\right)$. The total force on the charges within a volume $V$ can be found by integrating the tensor over the closed surface $S$ bounding that volume: $\mathbf{F}_{\text{total}} = \oint_S \mathbf{\sigma} \cdot \mathbf{n} dA$.

Here is the complete explanation:

The Maxwell stress tensor ($\mathbf{\sigma}$ or $\mathbf{T}$) is a mathematical object that quantifies the momentum flux of the electromagnetic field. It can be interpreted in two primary ways:

1. Stress (Force per Unit Area): Its components represent the force per unit area (stress) transmitted across an imaginary surface in the electromagnetic field.

   • The diagonal components ($\sigma_{ii}$) represent a tension (pull) or pressure (push) perpendicular to the surface.

   • The off-diagonal components ($\sigma_{ij}$ where $i \neq j$) represent a shear stress (tangential force) acting along the surface.

2. Momentum Density Flow: The tensor's divergence is related to the rate of change of electromagnetic momentum, making it central to the law of momentum conservation in electromagnetism.4

In the general case (including magnetic fields), the tensor components $\sigma_{ij}$ are defined as:

$\sigma_{ij} = \epsilon_0 \left(E_i E_j - \frac{1}{2} E^2 \delta_{ij}\right) + \frac{1}{\mu_0} \left(B_i B_j - \frac{1}{2} B^2 \delta_{ij}\right)$

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