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.
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$\complement\cdots$Counselor
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This is the central tool used in the proof. It allows us to convert a volume integral of a divergence into a surface integral over the enclosing surface. This is a powerful technique in vector calculus.
The identity $\nabla \cdot(\phi B)=(\nabla \phi) \cdot B+\phi(\nabla \cdot B)$ is the trick that makes the problem solvable. It links the dot product of the fields, $E \cdot B$, to a term that can be integrated using the divergence theorem.
The electric field lines are always perpendicular to the equipotential surfaces. This is a fundamental property of the electric potential.
The condition $\nabla \cdot B=0$ means there are no magnetic monopoles. The magnetic field lines always form closed loops without a beginning or end, and the net magnetic flux through any closed surface is zero.
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