🧄Conversion of Total Magnetic Force to a Surface Integral via the Maxwell Stress Tensor

Transforming the magnetic force integral is the conceptual shift from action-at-a-distance to the idea that the force is conveyed entirely by the Magnetic Stress Tensor ( $T ^{ i j }$ ) acting on the boundary surface. This rank-two tensor describes the momentum flux exerted by the magnetic field across that boundary, allowing the total force to be calculated simply by measuring the field B on the surface S. Physically, the tensor's components reveal the dual nature of these field mediated stresses: the off-diagonal terms ( $B_i B_j$ ) represent shear stresses that pull the surface diagonally, while the diagonal terms ( $\frac{1}{2} \delta_{i j} B^2$ ) represent a uniform outward pressure exerted perpendicular to the field lines.

Conversion of Total Magnetic Force to a Surface Integral via the Maxwell Stress Tensor | Proof and Derivationviadean on Notion