❓What is the Maxwell stress tensor and how is it related to the total electromagnetic force?

The Maxwell Stress Tensor ($\mathbf{T}$) is a mathematical tool in electromagnetism that fundamentally describes how the electromagnetic field exerts mechanical force and transfers momentum.

Here is a breakdown of its meaning and its relation to the total electromagnetic force:

The Maxwell stress tensor is a second-rank tensor (a 3x3 matrix) that represents the momentum flux of the electromagnetic field. In physical terms, it describes the force per unit area (stress) exerted across an imaginary surface by the field.

The components of the tensor describe:

• Diagonal elements ($T_{xx}, T_{yy}, T_{zz}$): Represent the pressure or tension exerted by the field perpendicular to the surface.

• Off-diagonal elements ($T_{xy}, T_{yz}$, etc.): Represent the shear stress—the tangential force exerted by the field along the surface.

Its components are defined in terms of the electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields:

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

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