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:

What is the Maxwell Stress Tensor ($\mathbf{T}$)?

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:

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

Relation to the Total Electromagnetic Force ($\mathbf{F}_{\text{field}}$)

The stress tensor provides an alternative and powerful way to calculate the total force that the field exerts on matter (or, equivalently, the negative force on the field itself, $\mathbf{F}_{\text{field}}$) inside a volume ($V$).

The total force on the electromagnetic field inside a volume is given by the general expression from the web page context:

$$ \mathbf{F}_{\text{field}} = \int_V \left(\mathbf{\nabla} \cdot \mathbf{T} - \frac{\partial \mathbf{g}}{\partial t}\right) d\tau $$

where $\mathbf{g} = \epsilon_0 \mu_0 (\mathbf{E} \times \mathbf{B})$ is the linear momentum density of the electromagnetic field (proportional to the Poynting vector).

1. General Case (Non-Static)

The term $\mathbf{\nabla} \cdot \mathbf{T}$ represents the rate at which momentum flows into a volume via the surface, and the term $\partial \mathbf{g} / \partial t$ represents the rate at which momentum accumulates in the field inside the volume. The total force on the field is the difference between these two rates.

2. Static Case ($\partial/\partial t = 0$)

In a static situation, where the fields and their momentum density are not changing with time ($\partial \mathbf{g}/\partial t = 0$), the force expression simplifies to:

$$ \mathbf{F}_{\text{field}} = \int_V (\mathbf{\nabla} \cdot \mathbf{T}) \, d\tau $$