Particles in Magnetic Fields

Did you know that magnetism allows the magnetron, and hence, the microwave oven to function? Before the invention of the microwave oven, however, magnetrons were actually in use for radar systems throughout World War Two! Microwaves are electromagnetic waves with wavelengths between a millimetre and a metre. They can be artificially generated by a device called a magnetron, which forces fast-moving electrons to travel in circular paths. It isn't easy to force something as small as electrons to move in a circle, but it is possible if we apply a magnetic field to them. When charged particles move through a magnetic field, the interaction creates a magnetic force on the particle that causes it to deflect. In this article, we explore the physics at work to produce the force on a charged particle, analyse how this affects the motion of particles in a magnetic field and finally look at the behaviour of some different types of particles.

As we know, electricity and magnetism are sections of physics that seem different but are related through the field of electromagnetism. This is a branch of physics that studies electromagnetic force, which is one of the four fundamental forces of nature and affects the interaction between electrically charged particles.

Magnetic force and magnetic field definitions

Experimentally, it was discovered that when electric charges are moving (that is, they have non-zero velocity), another contribution to the force the charged particles experienced was measured in addition to the electric force, which was already known. This mystery force was the electromagnetic force (also called magnetic force), caused by the movement of charge carriers through some externally applied magnetic field.

A simple experiment you can do at home to observe the effect of the magnetic force uses a strip of aluminium foil, a battery, and a horseshoe magnet. Initially, the magnet should produce no force on the foil strip, as it is made out of aluminium. However, if we create a circuit by attaching each end of the foil strip to the battery terminals, there are now electrons (charged particles) flowing through the foil. If we now bring the strip close to the magnetic field, it will deflect! This demonstrates that a magnetic force is generated by the movement of the charged particles through the magnetic field, as the force disappears when we disconnect the foil circuit and the particles stop moving.

Magnetic force is measured in newtons$\left(\mathrm{N}\right)$just like any other force. It is also important to note that the charged particle must be moving relative to the magnetic field to experience a magnetic force.

We must now uncover how magnets create this force, and for this, we need to discuss the magnetic field. The definition of the magnetic field is as follows.

A magnetic field is present at any point in space where a moving charged particle feels a force. The strength of a magnetic field is usually called the magnetic flux density or magnetic field strength and is given the symbol$B.$The unit of measurement of the magnetic field is the Tesla$\left(\mathrm{T}\right)$, which is equivalent to newtons-per-ampere-per-metre,${\mathrm{NA}}^{-1}{\mathrm{m}}^{-1}$.

Charged particles in a magnetic field

As we've learnt in the topic of electricity, the flow of positive electric charges constitutes a conventional electric current. Therefore, electric currents will experience a force when in a magnetic field. If we think about the electrons moving in circles in a magnetron, as we mentioned at the beginning of the article, how can we be sure that the electrons will move in that way, since they are not contained within a wire? It turns out that there is a relationship between the directions of the magnetic field, the magnetic force and the current. If we know the direction of any two of these quantities, we can find the direction of the third.

The left-hand rule

The way to determine the direction of the force that a moving charge will feel when it enters a magnetic field is by using Fleming's left-hand rule. The left-hand rule states that you should spread your thumb, index finger and middle finger so that they are all at right angles to each other (as shown in the diagram below).

1. Point your index finger in the direction of the magnetic field$B$, that is, north pole to south pole.
2. Your middle finger gives the direction of conventional current$I$(the movement of positive charge).
3. Your thumb will point in the direction of the force$F$on the particle.

The left-hand rule is used to determine the direction of the force felt by a charged particle when it moves through a magnetic field at right angles to it. Wikimedia Commons CC BY-SA 4.0

The force on charged particles in a magnetic field

The general law governing the behaviour of an electric charge in the presence of an electromagnetic field is known as the Lorentz force. The general expression also includes the effect of an external electric field, but here we will restrict it to situations where there is only a magnetic field present.

The expression for the force exerted on a charged particle moving perpendicularly through a magnetic field is given by:

$\begin{array}{rcl}\mathrm{Magentic}\mathrm{force}& =& \mathrm{charge}\times \mathrm{speed}\times \mathrm{magnetic}\mathrm{field}\mathrm{strength}\\ F& =& qvB\end{array}$

where$q$is the charge of the particle,$v$is the magnitude of its velocity (its speed), and$B$is the magnetic field strength.

This equation gives a clear indication that the particle must be moving$(v\ne 0)$for the magnetic force to be felt at all. We have a relationship in which the direction of the motion, force and field are all at right angles to each other and can be determined by using the left-hand rule.

The circular motion of charged particles in a magnetic field

We've seen that a charged particle will experience a force that is perpendicular to its direction of motion when it enters a magnetic field at right angles to the field. This means that the particle's direction of motion will change as the magnetic force deflects the particle. As the charged particle has now changed direction, this constitutes a change in direction of the current. If the field remains constant, but the current is changing direction, then the magnetic force generated must also constantly be changing direction! This makes for a confusing scenario but explains how a charged particle in a magnetic field can move along a circular path.