# Moving Charges in a Magnetic Field

Moving charges in a magnetic field can create problems for circuits. This is because electric charges are the building blocks of the electric field, and magnetic fields can affect them. So, what happens when we cancel out the electric field and just have a magnetic field? Let's explore the electromagnetic impact on a moving charge.

## Magnetic fields and electrical charges

Physics relies on fields, which are physical entities that exist in space and change over time. We use electric fields and magnetic fields to explain electric and magnetic phenomena. But here's the thing: these fields are not separate from each other. In fact, electric fields and magnetic fields are part of one common description based on charges. When charges are they generate only an electric field. But when they move, they create a magnetic field too. For our purposes, we'll focus on the magnetic field B, which is measured in Teslas (T). We'll also just consider point-like particles with a certain charge q, measured in Coulombs (C).

**Vector products**

Vector product is an operation between two vectors that produces another vector which is perpendicular to the two multiplied vectors. The magnitude of the resulting vector can be calculated by the formula shown below, where | | is the magnitude of a vector and θ is the angle formed between the two vectors. It is important to note that reversing the order of the vectors in a vector product results in a global minus sign.

In terms of geometry, the vector product has a significant consequence: the resulting vector is always perpendicular to the plane defined by the two original vectors. Additionally, if the angle between the two vectors is either zero or 180 degrees, the resulting vector will be zero.

To determine the direction of the resulting vector, we can use the right-hand rule, as illustrated in the image below.

**Moving charges in a magnetic field**

The Lorentz force is the general law that governs the behavior of an electric charge in the presence of an electromagnetic field. This law includes the effect of an external electric field, but for now, we will only consider situations where there is a magnetic field present.

The formula for the force that a magnetic field exerts on a moving electric charge is given by:

F = qv x B

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

The vector product in this formula means that the force exerted by a magnetic field on a moving charge is always perpendicular to both the direction of the magnetic field and the velocity of the charge. This also implies that charges that are not in motion will not be affected by the magnetic field. Additionally, if the charge is moving in the same direction as the magnetic field, it will not experience any force from the field.

## Charges moving in a uniform magnetic field

With the mathematical tools discussed in the previous section, we can create a phenomenological description of what happens when an electric charge moves through a region that has a magnetic field. By using the formula for the Lorentz force, we can analyze the dynamic trajectories of the particles and their energy.

The Lorentz force equation allows us to describe how a magnetic field affects the motion of a charged particle the direction and the force acting on a particle, we can determine the trajectory it will follow in the presence of a magnetic field.

Additionally, the Lorentz force equation can also be used to calculate the energy of charged particles in a magnetic field. When a charged particle moves perpendicular to a magnetic field, it experiences a force that causes it to travel in a circular path. This circular motion requires energy, which can be calculated using the formula for the work done by a force.

Overall, the mathematical tools provided by the Lorentz force equation allow us to study the effects of magnetic fields on the motion and energy of charged particles. This understanding is essential for a wide range of applications, from particle accelerators to magnetic resonance imaging.

**Mathematical description**

In this simplified scenario, we are considering a point-like particle with a charge q that is moving in a fixed direction at a constant velocity v. Initially, there is no magnetic field present in the region, but suddenly a constant magnetic field B is turned on. We will assume that the magnetic field is perpendicular to the velocity of the particle, which will result in the maximum vector product (with the sine function being equal to one).

The sudden introduction of the magnetic field will cause the particle to experience a force that is perpendicular to both the magnetic field and the velocity of the particle. This force will cause the particle to move in a circular path around the direction of the magnetic field.

The radius of the circular path can be calculated using the formula for the centripetal force:

F = ma = mv^2/r

where m is the mass of the particle and a is the acceleration due to the magnetic force.

Substituting the expression for the magnetic force from the Lorentz force equation, we get:

F = qvB

Equating the two expressions for F, we get:

qvB = mv^2/r

Solving for r, we get:

r = mv/qB

This expression shows that the radius of the circular path is inversely proportional to the strength of the magnetic field. Additionally, the speed of the particle will remain constant as it travels in a circular path, as the magnetic force only changes the direction of the velocity, not its magnitude.

Overall, this simplified scenario allows us to see how a sudden introduction of a magnetic field can cause a charged particle to move in a circular path. The radius of this path depends on the strength of the magnetic field and the properties of the particle.

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.

As we have seen, when a charged particle moves in a uniform magnetic field, it follows a circular trajectory with a constant speed. This means that the kinetic energy of the particle remains constant, as it is proportional to the square of the speed. However, the direction of the velocity changes due to the magnetic force, which means that the magnetic field is affecting the angular momentum of the particle.

When considering the interaction between magnets and metals, the situation is more complex, as the magnetic field is not uniform and the metal object is not a point-like particle. The interaction between the magnetic field and the electrons in the metal object can result in the generation of eddy currents, which can lead to energy losses due to the resistance of the metal.

Additionally, when a magnet attracts a metal object, the kinetic energy of the object increases as it moves towards the magnet, and it decreases as it moves away from it. This means that the energy of the system is changing due to the interaction between the magnetic field and the metal object.

Therefore, when studying the interaction between magnets and metals, it is important to take into account the non-uniformity of the magnetic field, the properties of the metal object, and the effects of eddy currents and energy losses.

## 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.

In particle accelerators, charged particles are first accelerated in a straight line using an electric field, and then directed into a region where there is a magnetic field. The magnetic field causes the particles to move in a circular path, with the radius of the path depending on the strength of the magnetic field and the speed of the particles.

By adjusting the strength of the magnetic field, it is possible to change the force exerted on the particles and thereby change their speed and direction. This allows for precise control over the movement of the particles and enables them to be accelerated in a circular circuit.

Particle accelerators are used in a variety of fields, including physics, medicine, and industry, to study the properties of particles, create new materials, and develop new technologies. The ability to control and manipulate the motion of charged particles using electric and magnetic fields is a key element of their design and operation.

To elaborate further, cyclotrons were an important advancement in the 20th century as they allowed for continuous acceleration of charged particles in a circular path, which was not possible with linear accelerators. This made them useful for a wide range of applications, including medical imaging and cancer treatment.

However, as particles approach speeds close to the speed of light, relativistic effects become more pronounced, and it becomes necessary to design more sophisticated devices that can take these effects into account. Synchrotrons are an example of such devices, and they are used for a variety of applications, including the production of short-lived radioactive isotopes for medical imaging and cancer treatment.

Overall, the study of the interaction between charged particles and magnetic fields has led to the development of many important technologies and has deepened our understanding of the fundamental forces of nature.

## Moving Charges in a Magnetic Field

**How does a charged particle move in a magnetic field?**

It moves according to the expression of the Lorentz force, which is perpendicular to the magnetic field and its velocity.

**Do magnets work in vacuum?**

Yes, the electromagnetic field and, in particular, the magnetic field do not need a medium to propagate.

**Why does a moving charge produce a magnetic field?**

Because a moving charge can be interpreted as an electric current, which are the main objects that create magnetic fields and are affected by them.

**What happens when electrons are immersed into a magnetic field?**

They will be deflected by the magnetic field according to the Lorentz force if their direction of movement is not parallel to the magnetic field.

**Does the energy of a moving charge in a magnetic field change?**

Its kinetic energy remains the same because it describes circular trajectories that do not modify the speed of the charge, only the direction of its velocity.