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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$\gg$Mathematical Structures Underlying Physical Laws
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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.
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.
Computing the Integral of a Static Electromagnetic Field
Computing the Integral of a Static Electromagnetic Field
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Proving the Cross Product Rules with the Levi-Civita Symbol
Proving the Epsilon-Delta Relation and the Bac-Cab Rule
Simplifying Levi-Civita and Kronecker Delta Identities
Why a Cube's Diagonal Angle Never Changes
How the Cross Product Relates to the Sine of an Angle
Finding the Shortest Distance and Proving Orthogonality for Skew Lines
A Study of Helical Trajectories and Vector Dynamics
The Power of Cross Products: A Visual Guide to Precessing Vectors
Divergence and Curl Analysis of Vector Fields
Unpacking Vector Identities: How to Apply Divergence and Curl Rules
Commutativity and Anti-symmetry in Vector Calculus Identities
Double Curl Identity Proof using the epsilon-delta Relation
The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
Surface Parametrisation and the Verification of the Gradient-Normal Relationship
Proof and Implications of a Vector Operator Identity
Conditions for a Scalar Field Identity
Solution and Proof for a Vector Identity and Divergence Problem
Kinematics and Vector Calculus of a Rotating Rigid Body
Work Done by a Non-Conservative Force and Conservative Force
The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
Calculating the Area of a Half-Sphere Using Cylindrical Coordinates
Divergence Theorem Analysis of a Vector Field with Power-Law Components
Total Mass in a Cube vs. a Sphere
Momentum of a Divergence-Free Fluid in a Cubic Domain
Total Mass Flux Through Cylindrical Surfaces
Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field
Computing the Integral of a Static Electromagnetic Field
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