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.
Magnetic Force Transmission via the Stress Tensor #audio
Transformation of Force: The derivation uses the Divergence Theorem and Maxwell's equations ( $\nabla \cdot \vec{B}=0$ and $\nabla \times \vec{B}=\mu_0 \vec{j}$ ) to transform the total magnetic force from a volume integral of force density ($\vec{G} \times \vec{B}$) into a surface integral. This allows calculating the force by only knowing the magnetic field $\vec{B}$ on the boundary $S$.
Physical Interpretation: The resulting rank-two tensor, the Magnetic Stress Tensor ( $T^{i j}$ ), describes the momentum flux exerted by the magnetic field across the boundary surface. This provides an elegant way to conceptualize force transmission: instead of action-at-a-distance, the force is due to stresses and tensions transmitted through the magnetic field itself.
Tensor Components: The components of the tensor show the interplay between field components:
$$ T^{i j}=\frac{1}{\mu_0}\left(B_i B_j-\frac{1}{2} \delta_{i j} B^2\right) $$
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