We are familiar with the idea that magnet has poles, and that “like” poles repel and “unlike” poles attract.

As with electric fields it can be useful to draw lines the reflect the magnetic field at a point. Field lines point from North to South.

The lack of magnetic monopoles means that magnetic field lines do not begin or end anywhere.

So in the case of a bar magnet we can see that the field lines that we can measure outside the magnet continue within it to close the loop.

If we put current through a wire we can determine that it produces a magnetic field.

The direction of the field can be determined by a right hand rule.

What is a magnetic element?

- it generates a magnetic field, which can attract or repel other magnetic elements
- a force is generated on it from another magnetic field

Either or both?

Newton's Third Law is universal!

We can give the length of the wire a direction and make it a vector $\vec{l}$.

The current is then defined to be positive when it flows in the direction of the length vector.

The force is then

$\vec{F}=I\vec{l}\times\vec{B}$

or in the diagram below

$F=IlB\sin\theta$

We can also chop the length up in to infinitesimal pieces which produce infinitesimal forces to accommodate a wire that changes it's direction with respect to a magnetic field, or a non-uniform magnetic field.

$d\vec{F}=I\,d\vec{l}\times\vec{B}$

The SI unit of magnetic field is the Tesla $\mathrm{T}$, named after Nikola Tesla.

Another unit for magnetic field is the Gauss $\mathrm{G}$, $1\mathrm{G}=10^{-4}\mathrm{T}$

The Lorentz equation combines the electric force and the magnetic force on an charged particle

$\vec{F}=q(\vec{E}+\vec{v}\times\vec{B})$

You can visualize these trajectories here.

A good way to represent the orientation dependence of the torque is to define a new vector quantity, the magnetic dipole moment

$\vec{\mu}=NI\vec{A}$

The direction of the vector can be determined by the right hand rule and we may now write the torque as

$\vec{\tau}=\vec{\mu}\times\vec{B}$

We can use the torque experienced by a loop in a magnetic field to produce a meter which measures current, ie. a Galvanometer.

In these devices a restoring force, usually provided by a torsion spring is required to bring the needle back to zero when no current is flowing.

The current flowing through the loop produces a torque

$\tau=NIAB\sin\theta$

where $\theta$ is the angle between the magnetic moment of the loop and the field.

This is balanced by the restoring force of the spring

$\tau=k\phi$

so that the deflection of the needle $\phi$ from it's zero position gives a measure of the current

$\phi=\frac{NIAB\sin\theta}{k}$

Ideally we would like to eliminate the dependence on $\theta$ and a combination of curved magnets and a iron core within the galvanometer coil are used so that the field is always as the same orientation to the wire coil.

Even with curved pole pieces we can see that if the coil in a galvanometer rotates past the point where the magnetic moment is aligned with the field the direction of the torque reverses, the result would be an oscillatory motion around this position.

In a motor we want to achieve torque which drives a continuous rotation. We can achieve this if the current would switch direction as we pass through the point where the torque is zero. With a DC source this is achieved using a commutator.

We can construct a simple motor in which torque is only generated during half the cycle (a simple form of commutation). All you need are:

- 1 AA battery
- 2 paperclips
- 1 strong neodymium magnet
- A length of magnet wire, which we will make into a coil

The key to making this motor work is that on one end only one side of the insulation is removed, so only in half a cycle will a current flow and the torque will always be in the same direction.

You can check out countless YouTube videos on how to make one. Here is an example.

Can we make a motor using a DC source without commutation?

This can be done with

- 1 AA Battery
- 1 strong neodymium magnet
- 1 screw
- 1 small length of wire

The device is a homopolar motor.

This is in fact the earliest kind of electric motor and was first demonstrated by Michael Faraday in 1821.

The device can be most easily understood by considering the force exerted by the magnet on the wire which we can see is into the page.

Again, there are countless videos. Here and here are examples.

We have seen that if there is a magnetic field applied at right angles to a current carrying wire, the charges moving in the conductor will experience a magnetic force.

In response, the charge carriers in the conductor will redistribute themselves to produce an internal electric field at right angles to the current and field so that the net force on a charge carrier is zero (or in other words, the system is in equilibrium).

The size of the field can be found by considering the Lorentz equation.

$\vec{F}=q(\vec{E}+\vec{v}\times\vec{B})$

The field required to achieve equilibrium is

$E_{H}=v_{d}B$

where we should recall that $v_{d}$ is the drift velocity of the carriers in the conductor.

The Hall field $E_{H}$ leads to a Hall emf across the sample proportional to it's thickness $\mathcal{E}_{H}=E_{ H}d=v_{d}Bd$

What is neat about the Hall effect is that the direction of the field depends on whether the carriers are positive or negative charges. See the explanation here.

A mass spectrometer separates particles according their mass to charge ratio.

Combining the Lorentz equation $\vec{F}=q(\vec{E}+\vec{v}\times\vec{B})$ with Newton's Second Law $\vec{F}=m\vec{a}$ gives us the equation of motion for charged particles in electromagnetic fields

$\frac{m}{q}\vec{a}=(\vec{E}+\vec{v}\times\vec{B})$

Various methods can be used to separate the particles according to their $\frac{m}{q}$ ratio.

One example is to have “crossed E and B fields” so that there is no net deflection: $qE = qvB$. This allows you to select a unique speed.

Then, one can measure the radius of the circle with a second magnetic field $B^\prime$: $qvB^\prime = mv^2/r$.