electrostatics and magnetostatics -凯发k8网页登录

electrostatics and magnetostatics

maxwell's equations describe electrodynamics as:

$\begin{matrix} \nabla \cdot d = ρ, \\ \nabla \cdot b = 0, \\ \nabla \times e = - \frac{\partial b}{\partial t}, \\ \nabla \times h = j \frac{\partial d}{\partial t} . \end{matrix}$

here, e and h are the electric and magnetic field intensities, d and b are the electric and magnetic flux densities, and ρ and j are the electric charge and current densities.

electrostatics

for electrostatic problems, maxwell's equations simplify to this form:

$\begin{array}{l} \nabla \cdot d = \nabla \cdot (ε e) = ρ, \\ \nabla \times e = 0, \end{array}$

where ε is the electrical permittivity of the material.

because the electric field e is the gradient of the electric potential v, $e = - \nabla v .$ , the first equation yields this pde:

$- \nabla \cdot (ε \nabla v) = ρ .$

for electrostatic problems, dirichlet boundary conditions specify the electric potential v on the boundary.

magnetostatics

for magnetostatic problems, maxwell's equations simplify to this form:

$\begin{array}{l} \nabla \cdot b = 0, \\ \nabla \times h = j \frac{\partial (ε e)}{\partial t} = j . \end{array}$

because $\nabla \cdot b = 0$ , there exists a magnetic vector potential a, such that $b = \nabla \times a$ . for non-ferromagnetic materials, $b = μ h$ , where µ is the magnetic permeability of the material. therefore,

$\begin{array}{l} h = μ^{- 1} \nabla \times a, \\ \nabla \times (μ^{- 1} \nabla \times a) = j . \end{array}$

using the identity

$\nabla \times (\nabla \times a) = \nabla (\nabla \cdot a) - \nabla^{2} a$

and the coulomb gauge $\nabla \cdot a = 0$ , simplify the equation for a in terms of j to this pde:

$- \nabla^{2} a = - \nabla \cdot \nabla a = μ j .$

for magnetostatic problems, dirichlet boundary conditions specify the magnetic potential a on the boundary.

magnetostatics with permanent magnets

in the case of a permanent magnet, the constitutive relation between b and h includes the magnetization m:

$b = μ h μ_{0} m .$

here, $μ = μ_{0} μ_{r}$ , where μ_r is the relative magnetic permeability of the material, and μ₀ is the vacuum permeability.

because $\nabla \cdot b = 0$ , there exists a magnetic vector potential a, such that $b = \nabla \times a$ . therefore,

$\begin{array}{l} h = \frac{1}{μ_{0} μ_{r}} b - \frac{1}{μ_{r}} m, \\ \nabla \times h = \nabla \times (\frac{1}{μ_{0} μ_{r}} \times a - \frac{1}{μ_{r}} m) = j . \end{array}$

the equation for a in terms of the current density j and magnetization m is

$\nabla \times (\frac{1}{μ_{r} μ_{0}} \nabla \times a) = j \nabla \times (\frac{1}{μ_{r}} m) .$

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