# Lecture 14 - Magnetic Fields

## Generating Electricity without Currents and Magnets

A Wimshurst machine allows the conversion of mechanical potential energy into a large electrical potential.

The amount of charge separated is not large, so while there is a very large potential difference between the two terminals the amount of electrical potential energy produced by the machine is not very large.

The large potential is possible because of the large capacitors.

You can find a nice explanation of how the machine works here

## Magnets

What we typically refer to as a magnet, i.e. a material that produces a magnetic field “permanently”, is in fact a ferromagnet. The name comes from the region where ferromagnetic stones were found in ancient Greek times, but magnets were also known in the same time period in India and China.

A compass is a freely suspended ferromagnet that can be used for navigation, or to determine the direction of a magnetic field.

## Poles

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

## What is a pole?

Poles always come in pairs, magnetic monopoles would be highly theoretically interesting, but have not been observed in experiment.

A magnetic monopole would be the magnetic equivalent of charge and would act as a source or sink of magnetic field.

We can think of a magnet as having a particular magnetization direction, and we can then understand why if we break a magnet we end up with the creation of another pair of poles.

In the context of the bar magnet we can consider the poles to describe the ends of a magnetic material.

## Magnetic Field Lines

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

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.

## Earth's magnetic field

The fact that a compass works demonstrates that the Earth has a magnetic field.

The magnetic field of the earth is also important in shielding the earth from cosmic radiation.

We should note that the magnetic North Pole is actually a magnetic south pole, and the magnetic South Pole is actually a magnetic north pole.

The Earth's magnetic poles move around, and in fact can flip their direction completely.

This might suggest to us that the magnetic field, unlike in our bar magnet, is due to some kind of dynamic process. In fact the Earth's magnetic field is due to electric currents in the outer liquid core.

Let us discuss how an electric current gives rise to a magnetic field.

## Magnetic field due to a current carrying wire

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.

Hans Chrisitan Oersted first noticed the effect on a compass due to a current during a lecture in 1820, which eventually led to fundamental new understanding about the connections between electricity and magnetism.

The basic idea can be tested with a compass, a wire and a battery, such as demonstrated here

One can visualize it better with this app.

## Magnetic field due to a current carrying loop

If we make the wire in to a loop we can again apply the right hand rule to determine the direction of the magnetic field.

One can visualize the field lines with this app.

We can compare the field distribution to the one we measured for the bar magnet, such as shown here.

## Force on a current carrying wire

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!

Having seen that a current carrying wire produces a magnetic field, we can now see whether a magnetic field exerts a force on a current carrying wire. In doing so we will be able to produce a definition of the magnetic field $\vec{B}$.

The direction of the force can be determined in a Jumping wire experiment.

Further experiment would reveal that the force on a wire is always perpendicular to both the current and the field, so we can see that we can use a right hand rule to determine the direction of the force.

So, the force is the cross-product of vector current and the vector magnetic field.

## Magnitude of magnetic field

Consider wire of which length $l$ lies within a magnetic field.

The force depends linearly on $l$ as well as the current $I$.

We can write an equation that contains this information as well as the right hand rule for the direction that we identified earlier.

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.

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

## Units of magnetic field

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

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

A Tesla is a fairly big magnetic field (some records can be found on the National Magnet Lab page.