Maxwell stress tensor
T_ij = ε₀(E_iE_j − ½δ_ij E²) + (1/μ₀)(B_iB_j − ½δ_ij B²). The 3×3 symmetric tensor whose divergence gives the mechanical force per unit volume the electromagnetic field exerts on charges and currents.
Definition
The Maxwell stress tensor T_ij encodes the mechanical effect of the electromagnetic field on matter as a momentum-flux density. Its components are T_ij = ε₀(E_iE_j − ½δ_ij E²) + (1/μ₀)(B_iB_j − ½δ_ij B²), symmetric in i and j. The divergence ∂_j T_ij gives the force per unit volume on charges and currents at a point; integrating ∮ T_ij n_j dA over a closed surface gives the total electromagnetic force on everything inside.
The components have a physical interpretation familiar from mechanical stress. The diagonal term T_zz = (ε₀/2)(E_z² − E_x² − E_y²) + (1/(2μ₀))(B_z² − B_x² − B_y²), for example, is a tension along the field lines when they run in the z-direction and a pressure perpendicular to them. Faraday had intuited this in the 1840s with his mental picture of field lines as "rubber bands under tension that repel each other sideways"; the Maxwell stress tensor is the rigorous mathematical statement of that image. Compute it for two parallel current-carrying wires and the tensor naturally reproduces the attractive/repulsive force. Compute it for a charge near a grounded plane and it reproduces the image-charge attraction.
The stress tensor also defines the electromagnetic momentum flux: since force is rate of change of momentum, a momentum-flux density is the right way to localise the field's mechanical action on matter. This leads directly to electromagnetic momentum density g = (1/c²)S = ε₀E×B, which carries its own implications — Abraham-Minkowski momentum debates, radiation pressure, recoil of a flashlight as it emits light. For a plane wave of intensity I hitting a perfect absorber, the pressure is I/c (not 2I/c — that's the formula for a perfect reflector, where incident and reflected momentum add). For sunlight at Earth, this is about 4.5 µPa for absorbers, 9 µPa for reflectors — small but nonzero, and the basis for solar-sail propulsion in space.