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 $\mathbf{T}$)

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:

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

How It Calculates Force on Charges

The Maxwell stress tensor is used to calculate the total electromagnetic force ($\mathbf{F}_{\text{EM}}$) acting on all charges and currents inside a given closed volume $V$.

The key principle is the Integral Form of the Momentum Conservation Law:

$$ \mathbf{F}_{\text{EM}} = \oint_S \mathbf{\sigma} \cdot \mathbf{n} dA - \epsilon_0 \mu_0 \frac{d}{dt} \int_V \mathbf{S} dV $$

Where:

• $\mathbf{F}_{\text{EM}}$ is the total Lorentz force on the charges within volume $V$.
• $\oint_S \mathbf{\sigma} \cdot \mathbf{n} dA$ is the surface integral. This term represents the total mechanical force transmitted across the boundary surface $S$ bounding the volume $V$. This is the "force due to stress" on the boundary.
• $\mathbf{n}$ is the outward unit normal vector to the surface $S$.
• $-\epsilon_0 \mu_0 \frac{d}{dt} \int_V \mathbf{S} dV$ is the volume integral. This term represents the rate of change of the total electromagnetic momentum stored inside the volume $V$ (where $\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}$ is the Poynting vector).

Static (Electrostatic) Simplification

In a static situation (like the one in the problem you shared), where charges are at rest and the magnetic field $\mathbf{B}$ is zero, the term involving the rate of change of momentum is also zero.