Lecture 12

Electric and magnetic dipoles compared

While electric and magnetic fields behave very differently in some respects, when it comes to dipoles, they have some major similarities.
Most importantly, electric fields align electric dipoles, and magnetic fields align magnetic dipoles
Once the dipoles are aligned, they have a big net effect (whether electric or magnetic)
As we’ve already seen, this can cause “stickiness” using electric dipoles (where dipoles within a material align to stick to a charged object).
In the magnetic case, this same effect is how fridge magnets and similar magnetic holders can ‘stick’ to some metals (even if the metal is not itself a permanent magnet).
The fields generated by dipoles also look similar between electric and magnetic cases, although with some important differences; see below.

Electric dipole

Magnetic dipole

	Electric	Magnetic
Origin	Charge separation, but net zero charge in volume	Current loop, but zero average current through volume
Dipole moment	$\vec{p} = q \vec{d}$	$\vec{\mu} = I \vec{A}$
Potential energy	$U = -\vec{p} \cdot \vec{E}$	$U = -\vec{\mu} \cdot \vec{B}$
Torque	$\vec{\tau} = \vec{p} \times \vec{E}$	$\vec{\tau} = \vec{\mu} \times \vec{B}$

Loops and coils

As discussed in the previous lecture, naturally-occuring magnets are in fact made out of lots of atomic-scale aligned magnetic dipoles;
Alternatively, we can generate an electrical current in the shape of a loop, and that will also have a magnetic dipole;
This will then also act rather like a bar magnet, but generated from an electrical current – a basic electromagnet;
If you stack lots of these loops on top of each other, or equivalently make a coil of wire, you can generate a magnetic field much stronger than a single loop.
Each time the wire goes round once is called a single ‘loop’ or ‘turn’.

Try moving the currents below to make something equivalent to making a 2D slice through a coil of wire. If you get it right, the magnetic field should end up looking very much like a bar magnet.

Note that when coils are densely stacked on top of each other, their magnetic fields add up. So, the magnetic dipole moment of a coil of $N$ loops with current $I$ and area $\vec{A}$ is

\[\vec{\mu} = N I \vec{A}.\]

Discussion point: what is the North end of the electromagnet that you made?

Magnetic flux

When calculating electric fields, we could either directly sum up the effect of all charges (lecture 3) or use Gauss’s law to calculate the flux of the electric field $\Phi_E$ through a surface (lecture 5). It turns out we can do something similar for magnetism. But it won’t turn out to be by using the magnetic flux $\Phi_B$. Let’s see why not.

By analogy with electric flux, we can define the magnetic flux $\Phi_B$ through a surface as

\[\Phi_B = \oiint \vec{B} \cdot {\rm d}\vec{A}.\]

This means we sum up the perpendicular component of the magnetic field over the surface.
This also translates into counting the number of field lines that pass through the surface, remembering that incoming field lines are negative and outgoing field lines are positive.
You can try drawing Gaussian surfaces on the example below, and moving the currents around.
You should find that the magnetic flux is always zero. If a fieldline enters a surface, it must also leave it.

The reason is that, unlike the electric field which has radial fieldlines around any charge, the magnetic field only ever has closed loops.
This means that the number of field lines entering a surface is always equal to the number leaving it.
Put another way, Gauss’s law for magnetism translates into $\Phi_B \propto n$ where $n$ is the number of magnetic charges¹ – but there are no magnetic charges, only electric ones! So we have Gauss’s law for magnetism:

\[\Phi_B=0\]

Just as with Gauss’s law for electromagnetism, proving this rigorously requires the divergence theorem which you will see next year.

The upshot is that magnetic flux, while a well-defined concept, is not practically useful in the way that electric flux is.

Ampère’s law

Ampère’s law captures the ‘circulation’ of the magnetic field, starting by summing up magnetic fields going around any loop of your choosing.
Let’s write down the law, then justify it afterwards. Here it is:
\[\oint \vec{B} \cdot {\rm d}\vec{l} = \mu_0 I_{\rm enc}.\]
where the integral is around a loop, and $I_{\rm enc}$ is the net current enclosed by the loop in the upwards direction according to the right-hand-rule. (We’ll illustrate this in the lecture.)
The left hand side imagines a circuit in space, and for each little step around that surface (${\rm d}{\vec {l}}$) calculates the magnetic field in the direction of the step ($\vec{B} \cdot {\rm d}\vec{l}$) then sums all these infinitessimal steps ($\oint$).
Note the two major differences compared to flux:
1. We integrate along a line that loops back to the start, not over a closed surface;
2. We take the magnetic field along the direction of the loop, not perpendicular to it.
The right hand side is just $\mu_0$ times the enclosed net current passing through the interior of the loop.

The mathematical origin of Ampère’s law is called Stokes’ theorem, and you will see this in full next year². But in the meantime you can understand how Ampère’s law works by considering the magnetic field around a straight wire with a current flowing up the positive $z$ axis:

From lecture 10 the magnetic field is:

\[\vec{B} = \frac{\mu_0 I}{2\pi r}\hat{\vec{\phi}}.\]

For a circle of radius r, the line element is ${\rm d} \vec{l} = \hat{\vec{\phi}}\, r \, {\rm d}\phi $, i.e. magnitude $r\, {\rm d} \phi$, pointing in the direction of increasing $\phi$.
Inserting this into the Stokes’ integral above, the integral becomes $\mu_0 I$ as expected (we will check this explicitly in the lecture).

The tricky bit is if you displace the circle so it’s not centred on the wire, or make the loop wobble around. Ampère’s law should still be valid, but how to show it? It turns out there is a rather convincing graphical method for doing this, which we will see in the lecture (and is Fig 28.17 in Y&F).

Applications of Ampère’s law

Like Gauss’s law, Amperé’s law is a powerful tool for calculating magnetic fields when there is a lot of symmetry.

The simplest case is a long straight wire.

The magnetic field must be symmetric around the wire, and must encircle it.
To get the strength of the magnetic field, Ampère’s law tells us that the circumference of the circle times the encircling magnetic field strength equals $\mu_0 I$.
But the circle has circumference $2 \pi r$, so the magnetic field strength must be $B = \frac{\mu_0 I}{2\pi r}$, as we found before.

This is just reversing the logic above, so it might seem like a cheat. But now consider the case of a coaxial cable, where there are two wires, one inside the other. This is used for high-speed internet cables, for example.

An electric current flows down the inner cable and back up the outer (or vice versa).
Drawing a circle centred on the cable, both wires are enclosed and by symmetry the magnetic field is the same at every point on the circle.
But the current in the inner wire is in the opposite direction to the current from the outer wire, so the total enclosed current is zero.
So the exterior magnetic field is also precisely zero.

This fact is closely related to how coaxial cables are highly resistant to electromagnetic interference³ (though to be careful, one needs to include the time-variations in the current which complicates things a bit).

Footnotes

Why there are electric charges but not magnetic charges (aka ‘magnetic monopoles’) is a fundamental question in physics that even now is only partially understood. Experiments like the LHC can search for new particles with a magnetic monopole, but so far have found no hint of one. ↩
As you might guess, Stokes’ theorem and the divergence theorem are closely related – from a modern perspective they are in fact both consequences of the incredibly powerful Stokes-Cartan theorem – but this is third or fourth year material, so don’t worry about it for now! ↩
In fact coaxial cable was invented to enable long-distance telegraph wires. ↩