# 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 $$\vec{F}=\oint\_S \vec{e}i T^{i j} d S\_j$, where $T^{i j}$$ 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.

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