Charged particles in a magnetic field.

Step 1 of 6

A moving charge

We saw that a wire carrying a current feels a force in a magnetic field, at right angles to the current and to the field. A current is charges on the move, so a single moving charge feels such a force too:

$F = qvB$⁠

Here $q$⁠ is the charge, $v$⁠ the speed and $B$⁠ the field, with the velocity at right angles to the field. Between the two coils there is a uniform field pointing up, and a gun at the centre of an airless glass chamber fires a Neon ion (Ne⁺) sideways. Its path glows.

Step 2 of 6

The radius

A force that is always at right angles to the motion is exactly what circular motion needs. So the magnetic force is the centripetal force:

$qvB = \frac{mv^2}{r} \quad\Rightarrow\quad r = \frac{mv}{qB}$$qvB =$$\frac{mv^2}{r} \quad\Rightarrow$$\quad r =$$\frac{mv}{qB}$⁠

Ion and Field strength are fixed. Predict what will happen to the radius when we double the Speed, then look.

Try it: after checking, set Speed back to 50 km/s. The circle shrinks back.

Back to the experiment ↑

Step 3 of 6

The time for a turn

How long does the ion take to complete a turn? It covers a circumference of $2\pi r$⁠ at speed $v$⁠. A bigger circle, but a faster ion too. Predict which of the two wins.

Step 4 of 6

A tighter circle

The radius depends on two things you can change here: Speed and Field strength. A stronger field pushes harder, and a slower ion is easier to bend. Only the Ion is fixed. Find a way to halve the radius.

Step 5 of 6

The charge-to-mass ratio

In the radius $r = \dfrac{mv}{qB}$⁠ the mass and the charge appear only as the ratio $\dfrac{m}{q}$⁠. So at a given speed and field, the circle reveals a particle’s charge-to-mass ratio, not either one on its own. Here the gun fires a Proton (H⁺), and everything is fixed. Its circle is very tight: a proton is very light for its charge.

The sign of the charge matters too: a negative ion, such as chloride, is pushed the other way and goes round the circle in the opposite direction.

Step 6 of 6

A mass spectrometer

At Helix the gun tips up a little. The speed along the field feels no force, so the circle stretches into a helix. At Mass spectrometer, the last stage of Build the spectrometer, a detector plate stands across from the gun, and the gun fires the element’s natural mix of isotopes. Each ion lands on the plate after half a turn, $2r$⁠ from the gun. Predict which will land farthest out.