6.3.2 Motion of charged particles

As we know from 6.3.1, a charged particle moving in a magnetic field will experience a force. We can demonstrate this using an electron deflection tube. The force on the beam of electrons can be predicted using Fleming's left hand rule. As the electrons enter the field they experience a downward force. The electrons change direction (due to the force) but the force (F) on each electron always remains perpendicular to its velocity. The speed of the electrons is constant as the force has no component in the direction of motion.

To find the force (F) acting on a charged particle of charge Q moving at a speed v at right angles to a uniform magnetic field of flux density B consider a section of conductor/beam of charged particles. The length L of this section is 'vt'. The force (F) on the conductor is given by:

F = B I L

F = B I (vt)

We are well aware that current is the rate of flow of charge. If there are N charged particles wach with a charge Q the current can be given by:

I = (NQ)/t

So the force acting on the conductor is given by:

F = B x (NQ/t) x vt = NBQv

So the force on each charged particle is:

F = (NBQv)/N = BQv

For an electron or proton  Q = e so F = Bev


Circular orbits
So we need to know a bit more about circular orbits of charged particles in a uniform magnetic field. Consider a charged particle of mass m and charge Q moving at right angles to a uniform magnetic field of flux density B. The particle will move in a circular path as the force acting on it is always perpendicular to its velocity. The centripetal force (F = (mv²) / r) on the particle is provided by the magnetic force (F = BQv):

BQv = (mv²) / r

r = mv/BQ

This equation shows that:

• faster-moving particles make a bigger circle
• more massive particles move in a bigger circle
• stronger magnetic fields make the particles move in a smaller circle
• particles with greater charge move in smaller circles

The velocity selector

We also need to know a bit about charged particles moving in a region occupied by both electric and magnetic fields. A velocity selector is a device that uses both electric and magnetic fields to select charged particles of specific velocity. It is a vital part of instruments such as mass spectrometers and some particle accelerators. It consists of two parallel horizontal plates connected to a power supply. They produce a uniform electric field of field strength E between the plates. A uniform magnetic field of flux density B is also applied perpendicular to the electric field. The charged particles travelling at different speeds to be sorted enter through a narrow slit Y. The electric and magnetic fields deflect them in opposite directions. Only for particles with a specific speed v will these deflections cancel so that they travel in a straight line and emerge from the second narrow slit Z. For an undeflected particle:

electric force = magnetic force

EQ = BQv

E = Bv

This means that the speed (v) depends only on electric field strength (E) and magnetic flux density (B).