Chapter 5 - Magnetostatics

Reference "Introduction to Electrodynamics" (5e) by David Griffiths.

Say instead of static electric charges (electrostatics), we start dealing with moving electric charges (electrodynamics).

Moving charges will generate a magnetic field around them. If the rate of moving charges (current $I$) is constant, we're working with magnetostatics. The $\vec{B}$-field for some current $I$ can be modeled by the right-hand rule:

Magnetic fields interact with one another. Using the Lorentz force law, $${\vec{F}}_{\text{mag}}=Q(\vec{v}\times \vec{B})$$ Or, in the presence of an electric field as well, $${\vec{F}}_{\text{mag}}=Q(\vec{E}+\vec{v}\times \vec{B})$$

Todo: cyclotron motion?

Note: magnetic fields do no work. $$d{W}_{\text{mag}}={\vec{F}}_{\text{mag}}\cdot d\vec{l}=Q(\vec{v}\times \vec{B})\cdot \vec{v}dt=0$$ The objects that generate the $B$-fields do work; but it's an apparently subtle distinction that the textbook does not expand on yet.

Current

Current is some charge per unit time through some cross-sectional area, defined such that $1\text{ampere}=1\text{Coulomb}/1\text{second}$.

If we had some line charge of $\lambda$ Coulombs traveling at velocity $v$, $I=\lambda v$. In cases where we're not just along a straight line, $\vec{v}$ is a vector and as such current is too: $$\vec{I}=\lambda \vec{v}$$ The force on this line of charge is $${\vec{F}}_{\text{mag}}=I\int d\vec{l}\times \vec{B}$$ for a constant current $I$.

Surface and Volume Currents

If we have some charges flowing across some $dl$ on a surface, or across some area $dA$ on a volume, we represent them with surface current density $\vec{K}$ and volume current density $\vec{J}$.

Similar to a line charge, if our surface has density $\sigma$ (or $\rho$ for a volume charge) and charges move at velocity $\vec{v}$: $$\vec{K}=\sigma \vec{v}\vec{J}=\rho \vec{v}$$ For some volume, the charge conservation equation (or continuity equation) says any charge flowing out of a volume means $$\begin{array}{rl}\nabla \cdot \vec{J}& =-\frac{\partial \rho }{\partial t}\\ & =0\text{(in magnetostatics)}\end{array}$$

Note: $\partial \rho /\partial t$ means change in charge density per change in time, meaning $\delta \vec{J}$. This is zero in magnetostatics.

Biot-Savart Law

The magnetic field of some steady-state line current is

$$\vec{B}(\vec{r})=\frac{{\mu }_0}{4\pi }I\int \frac{d{\vec{l}}^{\prime }\times \hat{R}}{R^2}$$

Note: $\vec{B}$ has units of newtons per ampere-meter (or Tesla): $1\text{T}=1\text{N/A}\cdot \text{m}$.

where $\vec{r}$ is the "test point" where you'd like to know the magnetic field, ${\vec{r}}^{\prime }$ points to some (temporary) infinitesimal charge line element $d{l}^{\prime }$, and $\vec{R}$ points from the charge element to the test point.

Note: the integration is along the current path.

${\mu }_0$ is the "permeability of free space": $${\mu }_0=4\pi \times 10^{-7}{\text{N/A}}^2$$

The B.S. law also works for surface and volume charges: $${\vec{B}}_{\text{surf}}(\vec{r})=\frac{{\mu }_0}{4\pi }K\int \frac{d{A}^{\prime }\times \hat{R}}{R^2}{\vec{B}}_{\text{vol}}(\vec{r})=\frac{{\mu }_0}{4\pi }J\int \frac{d{\tau }^{\prime }\times \hat{R}}{R^2}$$ where $d{A}^{\prime }=d{x}^{\prime }d{y}^{\prime }$ and $d{\tau }^{\prime }=d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }$, converting to polar or spherical as necessary.

Divergence and Curl of $\vec{B}$

The curl of any magnetic field is proportional to the current density (or contained current): $$\nabla \times \vec{B}={\mu }_0\vec{J}$$ where $\vec{J}$ is a volume current density. It's also related to total current $I$ by taking a surface integral bounding the volume: $$I_{\text{enc}}=\int \vec{J}\cdot d\vec{A}$$

This means stronger currents have higher-magnitude $\vec{B}$-fields, and stronger $\vec{B}$-fields enclose a higher current / current density within.

The divergence of a magnetic field is always zero. $$\nabla \cdot \vec{B}=0$$

A magnetic field will circle around a wire but will not expand outward.

Ampère's Law

For some circular magnetic field (circumference $2\pi r$) around a wire, the path integral of it is independent of radius: $$\oint \vec{B}\cdot d\vec{l}=\frac{{\mu }_{0}I}{2\pi }{\int }_0^{2\pi }d\varphi =\frac{{\mu }_{0}I}{2\pi }2\pi ={\mu }_{0}I$$

The radius of the path integral (circle, circ. $2\pi r$) increases at the same rate as the magnitude of the $B$-field decreases - i.e. it's independent of radius.

This is known as Ampère's Law: $$\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}$$ The current enclosed by some path integral along the $\vec{B}$-field is proportional to the enclosed current $I_{\text{enc}}$. It's like the magnetostatics equivalent to Gauss's law as it relates to Coulomb's law.

For a surface current along an infinite plane, the Ampèrean loop might be a rectangle perpendicular to the current.

Above, the $\hat{z}$ components of the Amp. loop aren't aligned with $\vec{B}$, so we only care about the $\hat{y}$ components - i.e. $$\begin{array}{rl}\oint \vec{B}\cdot d\vec{l}=B\oint d\vec{l}& ={\mu }_0{I}_{\text{enc}}\\ B(2l)& ={\mu }_{0}Kl\\ B& =\frac{{\mu }_{0}K}{2}\hat{y}(\text{for }z>0,\text{negative if }z<0)\end{array}$$

Note that Ampère's law only works for

1. Infinite straight lines / cylinders
2. Infinite planes
3. Infinite solenoids
4. Toroids (see Griffiths Ex. 5.10)

Magnetic Vector Potential $\vec{A}$

Magnetic vector potential $\vec{A}$ is the magnetostatics equivalent to electric potential $V$.

$\nabla \cdot \vec{B}=0$. If we instead write this as $\nabla \cdot (\nabla \times \vec{A})=0$, $$\vec{B}=\nabla \times \vec{A}$$ Griffiths has a further derivation of this in 5.4.1. Ultimately, we get to the magnetic Poisson's equation equivalent: $${\nabla }^2\vec{A}=-{\mu }_0\vec{J}$$ $$\vec{A}(\vec{r})=\frac{{\mu }_0}{4\pi }\int \frac{\vec{J}({\vec{r}}^{\prime })}{R}d{\tau }^{\prime }$$ We can pull out $J$ if constant. Use $\vec{K}({\vec{r}}^{\prime })d{a}^{\prime }$ or $Id{l}^{\prime }$ for surface and volume currents respectively.

The direction of $\vec{A}$ will almost always be the same as that of current.

Dipole Moment of $\vec{A}$

The explicit multipole expansion of $\vec{A}$ is in Griffiths 5.4.3. I've only included the dipole moment for brevity, and because the monopole moment has no proof of existence & higher-order terms are rarely helpful.

The dipole moment tends to dominate magnetic vector potential multipole expansions (no monopole given $\nabla \cdot \vec{B}=0$), and is represented as $${\vec{A}}_{\text{dip}}(\vec{r})=\frac{{\mu }_0}{4\pi }\frac{\vec{m}\times \hat{r}}{r^2}$$ where $m$ is the magnetic dipole moment $$m=I\int d\vec{a}=I\vec{a}(\vec{a}\text{ is vector area of Ampeˋran loop})$$ this is independent of origin, since $\vec{a}$ is centered at $\vec{r}=0$.