Tutorial 5 B - Magnetic Forces on Positively Charged Particles

Learning Objectives

To understand the effect of a magnetic field on positive charges

To learn how this is used in the mass spectrometer and the cyclotron.

Key Questions

Can a current be made form positive charges?

How does a mass spectrometer work?

What is a cyclotron?

The Mass Spectrometer

So far we have looked at electrons in magnetic fields.  A current can be formed using a stream of positive charges.  For example a stream of alpha particles from a source, if connected to a circuit will cause a small current.  The picture below shows a very simple battery based on an alpha source.

 The alpha source emits 106 alpha particles every second.  Calculate the current that it will produce.

The current is very small and would need a picoammeter to measure it.  But it shows that a current can be generated using positive particles.  The current is conventional, as it flows from positive to negative.  In a wire, the positive charges do not move; only the electrons move.

Positive charges are deflected by both magnetic fields and electric fields.  At GCSE, you will have seen that both alpha and beta particles are deflected by a magnetic field.  Gamma radiation is not deflected.

The mass spectrometer is an instrument that uses a stream of positively charged ions that interacts with a magnetic field.  It is the chemist's best friend.  However we will look at the physics of the instrument.  If you want to know about how it's used in chemistry, ask your chemistry tutor.  This link will give you the chemistry:

Here is a simplified diagram of the mass spectrometer:

A sample of material is placed into the machine and vaporised by intense heating in a vacuum.  The vaporised particles pass through into the ionisation chamber and electrons are knocked off as a result of collisions with electrons accelerated by the electron gun.  This makes the ions positive.  The electron gun anode attracts electrons and acts as an electron sink.

The positive ions are attracted to the ionisation cathode.  Most will hit it and lose their charge, but some will pass through the hole, and will travel at a certain speed, depending on their charge, the accelerating voltage and their mass.

The ions pass into the velocity selector.  This consists of crossed magnetic and electric fields that are arranged so that only those particles with a particular speed will pass through.

Once the ions have passed through the velocity selector, they are all travelling at the same speed.  Then they pass through another magnetic field.  The flux density of the magnetic field and the mass of the ion determine the radius of curvature.  If the mass is too high, the radius of the curvature will be larger than the radius of the tube, and the ions will hit the outside wall.  If the mass is smaller, then the radius of curvature will be less and the ion will hit the inside wall.  The ones that are just right will reach the detector.

To select the different ions we need to alter the magnetic flux density by changing the current.

The results from the detector look like this:

The read out in this case shows the relative abundance of three different isotopes a a particular element.  However it could be three different groups in an organic molecule.  Or it could be different elements that make up the sample of material.

Now let's have a look at the Physics.

Ionisation

This is done at a voltage of about 70 V.  Electrons are knocked off atoms to form positive ions.

Chemists use the first ionisation energy which is defined as the energy required to remove one electron from the outer shell of a mole of atoms.  The mole is defined as the same number of particles as there are in 12.0 g of Carbon-12 atoms.  It is often called Avogadro's number, and has the physics code LA.

LA = 6.02 × 1023 mol-1

Sodium has a first ionisation energy of 498 kJ mol-1.  Let's work out how much energy is needed per atom.  It's quite simple:

energy per atom (J) = ionisation energy (J mol-1) ÷ number of particles per mole (mol-1)

 Calculate the energy in joules and electron volts required to ionise one sodium atom.

Your answer to Question 14 will show that the ionisation voltage of 70 V is quite sufficient for the job.  Unlike excitation of atoms, which requires very specific energies, ionisation will happen if the incident electron energy is greater than the ionisation energy.

Ion Acceleration

The accelerating voltage for the positive ions is anywhere between 4000 V and 20 000 V.  We know that for electrons, all the energy given during acceleration is kinetic.  For positive ions a small fraction is turned into internal energy, but for our purposes we will take it that the ion energy is kinetic.  Therefore:

We can work out the mass of an atom by multiplying the number of nucleons (protons and neutrons) by the mass of a nucleon, which is 1.661 × 10-27 kg.  For sodium, there are 23 nucleons.  If we were to be completely accurate, we should take the mass of the electrons into account (11 × 9.11 × 10-31 kg = 1.00 × 10-29 kg).  For the mass of the ion, the extra mass of the electrons is the mass of 10 electrons (9.11 × 10-30 kg).  Why?  If we are working to two or three significant figures, it will make little difference, especially as we have neglected internal energy.

Do not confuse the mass of a nucleon (= 1.661 × 10-27 kg) with the mass of a proton (= 1.673 × 10-27 kg).  The difference isn't that much, but when we are doing calculations to more than two significant figures, the difference can be important.

 Show that the speed of a sodium ion accelerated by a potential difference of 4000 V is about 2 × 105 m s-1.

Velocity Selector

Now let's follow our sodium ion into the velocity selector.  This consists of plates that make an electric field that cross with a magnetic field.

We know that for an electric field:

F = Eq

We know that for a magnetic field:

F = Bqv

So we can equate the two to give:

Eq = Bqv

And it doesn't take a genius to see that the q terms cancel out:

E = Bv

 The sodium ion  passes at the speed you calculated in Question 15 through two plates that make an electric field.  The two plates have a potential difference of 4000 V between them.  The plates are separated by 5.0 cm.  Calculate the value of the magnetic field that is needed to keep the sodium ion on a horizontal path.

Note that charge does not come into it.  It is entirely possible that two particles will have exactly the same speed, but will have different masses and charges.  So how are they selected?

Deflection Magnet

The particles at the selected speed now go into a tube of a fixed radius.  They will be deflected by another magnetic field.  We know that the force applied to the charged particles is centripetal, and that the magnetic force and the centripetal force are equal:

Which gives us:

Or:

In this case, both mass and charge are involved.  So if we have a differing charge or mass, we will get a different path as shown below:

 Calculate the mass of a sodium ion. The path through the deflection magnet has a radius of 45 cm.  Using the value for speed you worked out in Question 15, calculate the value of the magnetic field needed to keep the ion on this path.

So far we have considered a sodium ion which has a single positive charge.  For Group II metals like magnesium, we need to take into account that it has a charge of +2e.  For Group III, the charge will be + 3e.

Detection

The ions striking the detector generates an electric current.  There is a computer that works out the particles are, having worked it out from the value of the magnetic fields, the accelerating voltage, and the electric field voltages.  The chemist gets a read out like this:

The read out is interpreted using the skill an experience of the chemist.  If you see such a read out and want to know what it means, ask a chemist.

Cyclotron

The cyclotron is a particle accelerator that uses the interaction between electric and magnetic fields. The machine’s main components are two D-shaped electrodes in an evacuated chamber, placed between the poles of a large electromagnet.

From the top it looks like this:

Notice that the beam of particles is not circular, but a spiral.  This is because the particles are being accelerated by the electric field between each D-shaped electrode (called a dee).  As their speed increases, so does the radius of the curved path.

If a particle of charge Q enters one of the dees with a speed v, it will move in a semi-circular path of radius r.  We know that:

This will rearrange to give us v:

We can work out form linear motion what time it takes for the charge to travel:

Therefore:

As the path is circular, we know that for one complete revolution:

Therefore we can substitute:

The r terms cancel out to give us:

Since f = 1/t, we can write:

and then rearrange to give our final result:

Click HERE to see an animation of the cyclotron by Stephen Lucas.

 A cyclotron is used to accelerate protons in a magnetic field of flux density 0.56 T.  Calculate the frequency that needs to be applied to the D-shaped electrodes.   Mass of proton = 1.673 × 10-27 kg Charge of a proton = 1.602 × 10-19 C