A static electromagnetic system where the boundary is an equipotential surface, the total integrated parallel component of the fields $(E \cdot B)$ within that volume must be zero. This result stems from the fact that the electric field can be expressed as the gradient of a potential, allowing the integrand to be rewritten as the divergence of the quantity $\phi B$ (since $B$ is solenoidal). By applying the Divergence Theorem, the volume integral reduces to the magnetic flux through the boundary surface; because that surface is equipotential, the potential factors out, and Gauss's Law for Magnetism dictates that the total magnetic flux through any closed surface is null.
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title Visualizing Electromagnetic Field Energy Geometry
Resulmation: Field Energy Density Visualization-static
: Field Energy Density Animator
IllustraDemo: Mapping Electromagnetic Energy With Divergence Theorem
Ex-Demo: The Geometry of Electromagnetic Potential and Field Energy
Narr-graphic: The Geometry of Energy in Static Electromagnetic Fields
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%% Proof and Derivation
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