# Finding e/m of an electron with a fine-beam tube

## The fine-beam tube apparatus

The fine-beam tube consists of a partially evacuated glass bulb, normally about the size of a small melon.

An electron gun accelerates electrons into a low-pressure gas, which glows as the electrons pass through so you can see the path they take.

## Increasing the accelerating voltage increases the speed of the electrons

If the electron gun voltage is increased then the electrons move faster. Their typical speed is of the order of 10 milion metres per second, say a few per cent of the speed of light.

## Why does the magnetic field make the electrons move in a circle?

The electrons keep *moving* because there is not much stopping them.

They move in a *circle* because the magnetic field continuously exerts a force on them. The force is always at right angles to their direction of motion, which is why it doesn’t speed them up or slow them down. All it does is change their direction continuously.

Where does this force come from?

Moving charges in a magnetic field experience a force. An electron is charged so it experiences a force. The direction of the force can be worked out using Fleming’s left-hand rule but it's always at right angles to the electron's direction of motion at each instant.

Note that there’s no middle of the field about which the electrons rotate. The electrons just feel this force which is always at right angles to the direction they're moving in and that’s why they move in a circle.

## The bigger the magnetic field the smaller the circle

What does the size of the force depend on and how does it effect the size of the circle?

The speed of the electrons depends only on the accelerating voltage. If we keep that constant then a bigger magnetic field means bigger force on the electrons, which seems sensible. If the force is bigger then it can pull the electrons into a tighter circle. Big force means small circle and vice-versa.

## Changing the speed of the electrons also changes the force on them

The higher the speed of the electrons the bigger the force on them.

But the circle gets bigger even though the force has increased. This seems to contradict what happens when we increased the magnetic field. Then bigger force meant *smaller* circle. Why is this?

Imagine the force was constant. If you make the electron go faster the force isn’t big enough to keep it in such a tight circle. So the circle would get bigger.

If we wanted to increase the speed and keep the size of the circle the same we’d have to increase the force somehow. But the size of the force does actually increase.

The problem is that it doesn’t increase enough.

The force needed to keep anything moving in a circle depends on the velocity squared. If you double velocity you need four times the force. But the force available only doubles. That’s why the circle gets bigger.

## The cyclotron frequency

There is one very useful effect that it’s worth noting in passing. The time to complete one revolution doesn’t depend on the speed of the electron if the magnetic field is kept constant. When the electron goes faster it has to go round in a bigger circle so the time taken stays the same.

The number of circles completed each second is called the ‘cyclotron frequency’ and is important in some types of particle accelerator.

## Finding the charge to mass ratio of an electron using the fine-beam tube

We can write down two equations which will help.

(i) centripetal force = magnetic force

and

(ii) energy gained by electron = energy lost by power supply

Note that the first equation says "Any force keeping something moving in a circle is called a centripetal force and in this case it's the magnetic force on the moving electrons". It doesn't mean there are two forces. It says that the centripetal force IS the magnetic force.

We end up with an expression for charge/mass for an electron in terms of things we can quite easily measure like the accelerating voltage, V, the radius of the circle, r, and the magnetic field strength, B. We can calculate the field strength, B, if we know enough information about the Helmholtz coils.

Thomson couldn’t use this method for various reasons but mainly because it was difficult to measure the accelerating voltage accurately.