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5: The Magnetostatic Field II

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    22824

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    Magnetostatic Boundary Value Problems for a Linear, Isotropic, Magnetic Material.

    • 5.1: Introduction- Sources in a Uniform Permeable Material
      This page covers magnetostatics equations, focusing on the interplay between magnetic field \(\vec B\), magnetic field strength \(\vec H\), and magnetization \(\vec M\) in linear, isotropic materials with permeability \(\mu\). It details key equations linking \(\vec B\) and free current density \(\vec J_f\), as well as the calculation of magnetic vector potential \(\vec A\) and its connection to \(\vec B\).
    • 5.2: Calculation of off-axis Fields
      This page explores the calculation of the magnetic field along the symmetry axis of an axially symmetric coil using the Biot-Savart law, focusing on the axial component and potential derived from Legendre polynomials. It details the derivation of the magnetic potential function and its coefficients, leads to the calculation of magnetic field components \( B_r \) and \( B_\theta \), and emphasizes the utility of digital computation for these tasks.
    • 5.3: A Discontinuity in the Permeability
      This page covers the intricacies of magnetic fields in materials with varying permeability, highlighting a stepwise approximation method and the use of magnetic scalar potential. It discusses boundary conditions, the continuity of magnetic fields, and specific potential functions for permeable shapes. The method of images is introduced for analyzing magnetic dipoles near permeable planes and interfaces, ensuring field continuity across boundaries.
    • 5.4: The Magnetostatic Field Energy
      This page covers the energy necessary for creating a magnetostatic field in linear isotropic materials, introducing energy density and deriving total energy through spatial integration tied to vector potential and current density. It utilizes Gauss’ and Stokes’ theorems to connect magnetic energy to circuit fluxes, culminating in a formulation for multiple circuits and highlighting similarities between electrostatic and magnetostatic systems.
    • 5.5: Inductance Coefficients
      This page examines the interplay between magnetic flux, currents in various circuits, and induction coefficients \(L_{MN}\) crucial for magnetostatics. It notes the symmetry of induction coefficients, resulting in \(N(N+1)/2\) independent values. The derivation of magnetic flux from currents is discussed, along with specific scenarios, such as the interaction of a primary coil with a smaller secondary coil.
    • 5.6: Forces on Magnetic Circuits
      This page explains the principles of magnetic flux, induced electromotive force, and energy conservation in circuits, particularly superconducting ones. It covers the forces on magnetic dipoles in inhomogeneous fields, deriving relevant equations for torque and mutual inductance. The discussion extends to magnetic energy in current-carrying loops and solenoids, detailing torque alignment with external fields and the effects of solenoid geometry on magnetic energy.
    • 5.7: The Maxwell Stress Tensor
      This page covers magnetic forces on current distributions in materials, similar to electrostatics. It presents the magnetic Maxwell stress tensor for calculating net forces using vector \(\vec T\)M. The force in linear magnetic materials is found through a surface integral of \(\vec T\)M, with its magnitude and direction influenced by magnetic field vectors \(\vec B\) and \(\vec H\), indicating tension when \(\vec B\) is normal to the surface and pressure when parallel.

    Thumbnail: Magnetic H-field inside and outside of a cylindrical bar magnet. (CC BY-SA 4.0; Geek3 via Wikipedia)

    This page titled 5: The Magnetostatic Field II is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by John F. Cochran and Bretislav Heinrich via source content that was edited to the style and standards of the LibreTexts platform.