❓Maxwell Stress Tensor and Surface Force

The two pages discuss the concept of the Maxwell Stress Tensor ($\mathbf{\sigma}$) in electrostatics. This mathematical object is used to calculate the force exerted by the electromagnetic field. The first page explains how the surface force on the field is calculated by integrating the stress tensor over an imaginary boundary surface ($\mathbf{F} = \int_{S} \mathbf{\sigma} \cdot \mathbf{n} , d \mathbf{A}$). It specifically details the component formulas for the force element ($\mathbf{dF}$), showing that the force is transmitted across the surface as a momentum flux or stress. The second page defines the Maxwell Stress Tensor itself, describing it as a symmetric second-order tensor whose components represent the force per unit area (stress) transmitted across a surface, including diagonal components for tension/pressure and off-diagonal components for shear stress. It further clarifies that in electrostatics, the total force on the charges inside a volume is determined entirely by this same surface integral ($\mathbf{F}{\text{EM}} = \oint{S} \mathbf{\sigma} \cdot \mathbf{n} , d \mathbf{A}$), effectively viewing the field as a medium under stress that transmits force to the charges.

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