Moving Charges in a Magnetic Field

When we build circuits, it is never a good idea to use magnets next to them. The reason for this is that the basic units of the electric field are electric charges, which are affected by magnetic fields. Let’s take a look at the electromagnetic influence on an electric charge to see what happens when we set the electric field to zero.

Magnetic fields and electrical charges

Modern physics is based on the use of fields, which are time-dependent physical entities that extend in space. For electric phenomena, we use electric fields and the laws that govern their behaviour, while for magnetic phenomena, we use magnetic fields and the laws that govern their behaviour.

It is important to note that electric fields and magnetic fields are not independent of each other. Historically, it was a difficult process to understand that both physical fields are part of one common description that is based on charges that, if they are static, generate only an electric field but, upon moving, also generate a magnetic one.

Here, we only need to consider the magnetic field B as a time-and-space-dependent vector field. Magnetic fields are measured in Teslas (T). We will also only consider point-like particles with a certain value of a charge q that is measured in Coulombs (C).

Vector products

A vector product is an operation between two vectors that yields another vector. The resultant vector is perpendicular to the two multiplied vectors and has a module that can be computed as:

\[|\vec{v} \times \vec{B}| = |\vec{v}| \cdot |\vec{B}| \cdot \sin (\theta_{vB})\]

Here, | | indicates the module of a vector, and the angle is the angle formed between the vectors. The vector product has the following property: reversing the order of vectors in a vector product amounts to a global minus sign, i.e.:

\[\vec{v} \times \vec{B} = -\vec{B} \times \vec{v}\]

The main consequence of considering vector products is that the resulting vector is perpendicular to the plane defined by the other two vectors and that if their angle is zero or 180º, the vector product is the zero vector.

A useful way to determine the direction of the resulting vector is to use the right-hand rule, which is depicted in the image below.

Moving charges 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 ourselves to situations where there is only a magnetic field present.

The expression for the force exerted by a magnetic field on a moving electric charge is:

\[\vec{F} = q \cdot \vec{v} \times \vec{B}\]

Here, v is the vector velocity, and the product between the velocity and the magnetic field is a vector product.

The vector product implies that the force exerted by a magnetic field on a moving charge is perpendicular to the direction of the field and the velocity of the charge. It also implies that charges that are not moving do not ‘see’ the magnetic field since they are not affected by it. Furthermore, if the charge is moving in the same direction as the magnetic field, it will not feel its influence.

Charges moving in a uniform magnetic field

Using the mathematical tools of the previous section, we can provide a phenomenological description of what happens when an electric charge is moving in a region where there is a magnetic field. From the formula of the Lorentz force, we can study the dynamical trajectories as well as the energy of the particles.

Mathematical description

We now restrict ourselves to the case where the magnetic field has a constant fixed value B that does not depend on space or time. We also restrict ourselves to the case of a constant initial velocity v.

Our setting is the following: a point-like particle with a charge q is travelling in a fixed direction at constant velocity. Without loss of generality, we can consider this direction to be the x-axis. The particle is travelling in a region where there is no magnetic field until it is suddenly turned on. We will consider the magnetic field to be perpendicular to the velocity, so we have a maximum vector from the vector product (with the sine function being equal to one).

As soon as the magnetic field is turned on, the magnetic force makes the particle turn in the direction determined by the Lorentz force. In this case, according to the formula, the index finger points in the direction of the movement of the charge, while the middle finger is pointing in the direction of the magnetic field. Since the velocity changes due to the action of this force, the force now acts in a different direction. If you slowly turn the fingers with the right-hand rule, you realise that the particle is bound to describe a circle, as the direction of the force is constantly changing.

For this kind of setup, there is a convention for the direction of the magnetic field, according to which we use crosses to denote a magnetic field entering the page and circles for a magnetic field that exits it while being directed towards the observer.

What are cyclotrons?

We finally consider an application of the effect we have just studied: cyclotrons, which are accelerators of particles that are based on the Lorentz force.

Essentially, particles are first accelerated thanks to an electric field (in a straight line) and then arrive in a region where there is a magnetic field, which causes them to describe a circular motion. The intensity of the magnetic field can be changed in order to exert a higher force on the particle and change its speed and velocity. This allows accelerating particles in a circular circuit.

Cyclotrons were an advancement in the 20th century as only linear accelerators had been used before, which did not allow to keep the acceleration going. On the other hand, when they reach speeds close to the speed of light, experiments suggest that we must look for better-designed devices that take into account radiative effects as well as relativistic ones. These improved devices are known as synchrotrons, which are used, for instance, in the production of short-lived radioactive isotopes.