📢From Currents to Surfaces: Unlocking the Total Magnetic Force with the Maxwell Stress Tensor

The derivation outlines the process of converting the total magnetic force ( $\vec{F}$ ) acting on a volume V, initially expressed as a volume integral dependent on the internal current density ( $\vec{F}=\int_V(\jmath \times B) d V$ ), into a surface integral involving only the magnetic field ( $\vec{B}$ ) on the boundary surface S. This conversion requires that the magnetic field satisfies the Maxwell conditions $\nabla \cdot \vec{B}=0$ and $\nabla \times \vec{B}=\mu_0 \vec{\jmath}$. By substituting $\vec{\jmath}$ using Maxwell's equation, applying vector identities facilitated by $\nabla \cdot B=0$, and manipulating the components using the Kronecker delta ( $\delta_{i j}$), the force component $F_i$ is recast as the divergence of a tensor field. Finally, the Divergence Theorem (Gauss's Theorem) is applied to transform this volume integral into the required surface integral, yielding the result is the rank two Magnetic Stress Tensor with components $T^{i j}=\frac{1}{\mu_0}\left(B_i B_j-\frac{1}{2} \delta{i j} B^2\right)$, describing the momentum flux per unit area across the surface S.

From Currents to Surfaces: Unlocking the Total Magnetic Force with the Maxwell Stress Tensor | FAQsviadean on Notion