Maxwell stress tensor

In SI units, one common convention for the vacuum Maxwell stress tensor has components Tᵢⱼ = ε₀(EᵢEⱼ − ½δᵢⱼE²) + (1/μ₀)(BᵢBⱼ − ½δᵢⱼB²). Here E and B are the electric and magnetic fields, ε₀ and μ₀ are the vacuum permittivity and permeability, and δᵢⱼ is the identity tensor. The tensor packages directional electromagnetic stresses and momentum flux into a quantity that can be integrated over a surface. Equivalent sign or index conventions occur, so a calculation must keep the chosen convention consistent with its force equation.

For a closed surface S in vacuum surrounding a body, the surface integral of the tensor acting on the outward normal gives the net electromagnetic force in the static-field case: F = ∮ₛ T·n dA, under the convention above. The surface need not coincide with the object's physical boundary; it can be selected where the fields are easier to evaluate, provided it encloses the same object and the intervening region satisfies the assumptions of the calculation. For time-dependent fields, the change in electromagnetic momentum within the enclosed volume also enters the momentum balance, so the surface flux alone need not equal the instantaneous mechanical force. Torque can similarly be obtained by taking the moment of the surface traction about a chosen origin.

The tensor method turns a force calculation into a surface integral of field quantities. It is useful when the fields on a closed surface are easier to find than the detailed forces on every charge or current inside it. In electromechanical analysis it can help determine forces, torques and boundary conditions; modern modelling work also develops generalized forms for semi-analytic calculations involving permanent magnets, coils and soft iron. The quality of a numerical result depends on field accuracy, surface selection and numerical integration.

The vacuum formulation is widely used to express electromagnetic momentum balance, but the physical interpretation of stress tensors in matter is not always straightforward. In particular, assigning electromagnetic and mechanical momentum within material media has prompted competing formulations and ongoing scholarly discussion. A tensor-based force calculation should therefore state its assumptions—such as vacuum surrounding the body, constitutive model and whether fields are static—and distinguish electromagnetic momentum flux from the material's ordinary mechanical stress. Different conventions can also change the apparent sign of the tensor while leaving consistent physical predictions unchanged.