# What is a magnetic field

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

Magnetic field ($\boldsymbol{B}$) is force per charge per speed. It is a vector (strictly, a pseudovector), measured in units of N.s/C.m = N/A.m = T (tesla).

The force from a magnetic field $\boldsymbol{B}$ on a charge $q$ with velocity $\boldsymbol{v}$ is $q\,(\boldsymbol{v}\times\boldsymbol{B})$.

The force from a magnetic field $\boldsymbol{B}$ on a current $I$ in a straight wire with vector length $\boldsymbol{l}$ is the Laplace force, $I\,(\boldsymbol{l}\times\boldsymbol{B})$.

A magnetic field is produced (induced) by moving electric charge, such as a current.

An endless electric solenoid with current $I$ and constant pitch (turns per length) $n$ has a magnetic moment density $\boldsymbol{h}$ along the solenoid, with $|\boldsymbol{h}|\ =\ nI$, which produces a magnetic field along the middle, $\boldsymbol{B}\ =\ \mu_o\boldsymbol{h}$.

In empty space, an endless electric solenoidal field with magnetic moment density field $\boldsymbol{H}$ is the same as a magnetic field $\boldsymbol{B}\ =\ \mu_o\boldsymbol{H}$.

Magnetic moment density ($\boldsymbol{H}$) is a vector, current times pitch, measured in amp-turns per metre (A/m).

$\mu_o$ is a universal constant which should really be $1\ N/A^2$, but is actually defined as $4\pi\ 10^{-7}\ N/A^2$, so as to make the amp (A) a useful everyday unit.

Equations

Lorentz force: $q\,(\boldsymbol{E}\ +\ \boldsymbol{v}\times\boldsymbol{B})$.

Laplace force: $I\,(\boldsymbol{l}\times\boldsymbol{B})$

Force on a moment: $\nabla(\boldsymbol{m}\cdot\boldsymbol{B})$

Biot-Savart law:
$$\boldsymbol{B} = \frac{\mu_oI}{4\pi} \int \frac{d\boldsymbol{l} \times \hat{\boldsymbol{r}}}{r^2}$$
Gauss' Law for Magnetism: $\nabla\cdot\mathbf{B}\ =\ 0$

Faraday's Law: $\nabla\times\mathbf{E}\ =\ - \frac{\partial\mathbf{B}}{\partial t}$

$\boldsymbol{B} = \mu_o(\boldsymbol{H} + \boldsymbol{M})$

Extended explanation

Electromagnetic field:

The magnetic field $\boldsymbol{B}$ is a vector, but it transforms (between observers with different velocities) as three of the six components of a 2-form, the electromagnetic field, $(\boldsymbol{E};\boldsymbol{B})$.

Lorentz force:

The magnetic force is part of the whole Lorentz force, $q\,(\boldsymbol{E}\ +\ \boldsymbol{v}\times\boldsymbol{B})$.

The magnetic force on a stationary charge ($\boldsymbol{v}\ =\ 0$) is zero.

(Unless that charge has a magnetic moment, see next section.)

On a moving charge, it changes the direction but not the speed … so (in a constant magnetic field) the charge moves with constant speed in a circle … its kinetic energy stays the same … the field does no work on it.

Force on a current:

The Lorentz force on an uncharged stationary conductor (such as a wire) is zero, unless a current is flowing through it: that means that the (positively charged) nuclei are stationary, but some of the (negatively-charged) electrons are moving, and therefore are affected by a magnetic force, $(\sum\ q\boldsymbol{v})\times\boldsymbol{B}$.

$\sum\ q\boldsymbol{v}$ for the moving electrons is the sum of charge times distance per time, = distance times charge per time, = distance times current.

Accordingly, the force on a current-carrying stationary straight wire of vector length $\boldsymbol{l}$ is the Laplace force $I\ (\boldsymbol{l}\times\boldsymbol{B})$.

Force on a moment:

Even a stationary electron has a magnetic moment, as if it was spinning with a finite radius and angular speed.

The force from a magnetic field $\boldsymbol{M}$ on a magnetic moment $\boldsymbol{m}$ is $\nabla(\boldsymbol{m}\cdot\boldsymbol{B})$.

This is the force which enables a magnetic field to attract stationary magnetisable material in which the electron moments are not random.

Force from a current:

The Biot-Savart law states that the magnetic field at point $\boldsymbol{r}$ induced by a current I in a wire whose line element is dl is $\boldsymbol{B} = (\mu_oI/4\pi) \int (d\boldsymbol{l} \times \hat{\boldsymbol{r}})/r^2$.

More convenient is the law for the magnetic field along the middle of a solenoid of constant pitch $n$ and current $I$: $|\boldsymbol{B}|\ =\ \mu_onI$.

Magnetic moment density (dipole moment density):

The magnetic moment of one turn of a current $A$ enclosing a planar area $A$ is the vector $\boldsymbol{m}$ along the normal (perpendicular) direction, with $|\boldsymbol{m}|\ =\ IA$.

So the magnetic moment density of a solenoid with a current $I$ and with pitch $n$ (in turns per metre) is the vector $\boldsymbol{h}$ along the solenoid, of magnitude $|\boldsymbol{h}|\ =\ IA$ times number of turns over volume, $|\boldsymbol{h}|\ =\ nI$.

Magnetic moment density is a vector, measured in units of amp-turns per metre $(A/m)$.

A magnetised material also has magnetic moment density, from the loops of "bound current" constituting its magnetisation.

Two ways of measuring a magnetic field:

A magnetic field can be measured according to its effect, or its cause.

Its effect comes from the Lorentz force: force per charge per speed.

So it can be measured in units of N/C(m/s), or N/(C/s).m, or N/A.m (newton per amp-metre). This is defined as the tesla (T).

Its cause, in most cases, is loops of current (artificially, from solenoids, or naturally, from "bound current" in magnetised material), and its strength is proportional to the magnetic moment density of such solenoids or material.

So it can be measured in the same units as magnetic moment density: amp-turns per metre (A/m).

Historically, the B field has always been measured in tesla, and the H and M fields in amp-turns per metre: but there is no reason why they cannot be measured in the same units.

What is µ0?

µo is the conversion factor between tesla ($T\ =\ N/A.m$) and amp-turns per metre ($A/m$): so it has units of $N/A^2$.

Why isn't µo = 1 N/A2 (so that it needn't be mentioned)?

well, it would be , buuuut

i] in SI units, a factor of 4π keeps cropping up! … so we multiply by 4π

ii] that would make the amp that current which in a pair of wires a metre apart would produce a force between them of 2 N/m …

which would make most electrical appliances run on micro-amps! :yuck: …

so, for practical convenience only, we make µo 107 smaller, and the amp 107 larger!

(so the amp is that current which in a pair of wires a metre apart would produce a force between them of 2 10-7 N/m, and µo is 4π 10-7 N/A2 (= 4π 10-7 H/m))

(for historical details, see http://en.wikipedia.org/wiki/Magnetic_constant)

How can it be appropriate to say that empty space contains a magnetic moment density, varying from point to point, when there are no actual loops of current anywhere near?

Because any magnetic field can be replaced by an identical solenoidal field, as follows:

Let's define a solenoidal field as a region R of space with a "honeycomb" of thin hexagonal solenoids (they needn't be hexagonal: but that makes them fit nicely ), each with a (different) current Ii, and a (different) pitch, ni (pitch is turns per length).

The solenoids aren't straight, they can be curved into any shape.

That causes ("induces") the whole region R to be filled with a magnetic field, of "Lorentzian" strength µoniIi, = µohi, inside each solenoid, where hi is the magnetic moment density of each solenoid, measured in amp-turns per metre.

Now consider any B field in any region R.

We can fill R with an imaginary honeycomb of solenoids whose sides follow the B field lines (ie lines of constant |B|, and whose tangent at each point is parallel to the B field at that point), and whose current or pitch (or both) are adjusted so that the solenoidal field equals the B field along the centre line of each solenoid …

and by making the number of solenoids large enough (ie, the diameters small enough), we can make the solenoidal field match the whole B field to any required degree of accuracy.

In other words: in the limit, any actual B field can be replaced by a purely solenoidal field, which is naturally described in amp-turns per metre.

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