🧄Computing the Integral of a Static Electromagnetic Field

By applying the divergence theorem in conjunction with the vector identity $\nabla \cdot( \phi B )=( \nabla \phi ) \cdot B + \phi ( \nabla \cdot B )$ and the physical principles of electric fields being perpendicular to equipotential surfaces, and magnetic fields being divergence-free, it can be proven that the volume integral of $\overrightarrow{ E } \cdot \overrightarrow{ B }$ inside a closed surface where the potential $\phi$ is constant is equal to zero.

🎬Computing the Integral of a Static Electromagnetic Field

The demonstration of how the divergence theorem and boundary conditions can be applied to prove that the volume integral of the dot product of an electric field (radial, emanating from an equipotential surface) and a divergence-free magnetic field (tangential) inside a closed surface equals zero. This is visually represented through a sequence of frames, showcasing the setup, the fields, the integrand mapping, and the final result.

🖊️Mathematical Proof

Computing the Integral of a Static Electromagnetic Field | Proof and Derivationviadean on Notion