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

Context: Graduate
Thread starter Charles Link
Start date Aug 12, 2019
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Curl Curl operator Magnetostatics Operator

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SUMMARY

Maxwell’s equations in magnetostatics, specifically the equation \(\nabla \times \vec{B}=\mu_o \vec{J}_{total}+\mu_o \epsilon_o \dot{\vec{E}}\) with \(\dot{\vec{E}}=0\), are essential for solving magnetostatic problems. The discussion highlights the importance of the magnetic pole model equation \(\vec{B}=\mu_o \vec{H}+ \mu_o \vec{M}\) and addresses the lack of comprehensive resources on these topics. Two specific problems are analyzed: one involving an integral expression for \(\vec{H}\) and another deriving the magnetomotive force (MMF) equation, which is crucial for electrical engineers working with transformers.

PREREQUISITES

Understanding of Maxwell's Equations in differential form
Familiarity with the concepts of magnetostatic fields
Knowledge of vector calculus, particularly the curl operator
Basic principles of electromagnetism and magnetic materials

NEXT STEPS

Study the derivation and applications of the magnetomotive force (MMF) equation
Explore the integral expressions for magnetic fields in magnetostatics
Learn about the role of magnetic materials in electromagnetism
Investigate advanced vector calculus techniques used in electromagnetism

USEFUL FOR

Electrical engineers, physicists, and students studying electromagnetism who seek a deeper understanding of magnetostatics and its applications in transformer design and analysis.

Aug 12, 2019

Charles Link

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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...

Last edited by a moderator: Jun 11, 2022