The Maxwell Stress Tensor's physical interpretation is that it describes the momentum flux per unit area across the boundary surface $S$ due to the magnetic field. By transforming the force calculation from a volume integral (which required knowing the current distribution inside) to a surface integral of this tensor, the formula shows that the total force on the volume is entirely determined by the stresses-the pressures and tensions-exerted by the magnetic field on the boundary surface. Essentially, the field itself is treated as a medium transmitting mechanical momentum.
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