Maxwell’s Equations in Magnetostatics and Solving with the Curl Operator

[Charles Link]

Introduction:
Maxwell’s equation in differential form $\nabla \times \vec{B}=\mu_o \vec{J}_{total}+\mu_o \epsilon_o \dot{\vec{E}}$  with $\dot{\vec{E}}=0$ comes up quite frequently in magnetostatic problems.  In addition, the equation from the magnetic pole model $\vec{B}=\mu_o \vec{H}+ \mu_o \vec{M}$ comes up quite often as well.  It seems the textbooks are somewhat lacking in a thorough treatment of the use of these two equations,  and the mathematical operations that can be used to generate solutions.   In this Insights article,  we will attempt to fill that gap.
In this paper,  we will consider two rather different problems,  which both employ the vector $\vec{H}$.  The first involves an integral expression for $\vec{H}$.  The second involves a derivation of the magnetomotive force (MMF) equation.  The EE’s often use this equation in working with transformers.  Here we will show that this MMF equation arises from an alternate form of Maxwell’s/Ampere’s...

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[Vailanor]

The first problem we will consider is an integral expression for vector H. This can be derived by taking the curl of both sides of the equation $\nabla \times \vec{B}=\mu_o \vec{J}_{total}+\mu_o \epsilon_o \dot{\vec{E}}$.On the left hand side we obtain:$\nabla \times (\nabla \times \vec{B}) = \nabla \times (\mu_o \vec{J}_{total}+\mu_o \epsilon_o \dot{\vec{E}})$Using the vector identity, $\nabla \times (\nabla \times \vec{A}) = \nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$, and then equating with the right hand side, we obtain the following expression:$\nabla (\nabla \cdot \vec{B})-\nabla^2 \vec{B}=\nabla \cdot (\mu_o \vec{J}_{total}+\mu_o \epsilon_o \dot{\vec{E}})$By making the substitution $\vec{H}=\frac{\vec{B}}{\mu_o}$, we can rewrite this expression as$\nabla (\nabla \cdot \vec{H})-\nabla^2 \vec{H}=\nabla \cdot (\vec{J}_{total}+\epsilon_o \dot{\vec{E}})$Finally, if we make the further substitution $\vec{J}_{total}=-\nabla \phi$, we obtain the final form of our integral expression for vector H:## \nabla (\nabla \cdot \vec{H})-\nabla^2 \vec{H}=\nabla \phi-\nabla \cdot (\epsilon_o \dot{\vec{E}})