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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…

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# Moving Charges in a Magnetic Field

Moving Charges in a Magnetic Field
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Nie wieder prokastinieren mit unseren Lernerinnerungen. 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. Figure 1. Right hand rule.

### 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). Figure 2. Charges with opposite signs approaching a region with a magnetic field going into the page., Wikimedia Commons

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. Figure 3. Trajectory of a charge in a magnetic field going into the page, Wikimedia Commons

We have seen that moving charges in a uniform magnetic field describe circular trajectories. The general theory for circular motion states that the speed of the object describing it does not change, while its velocity (direction) does, which is exactly what happens with the Lorentz force.

This affects the energy of the particle since the kinetic energy is proportional to the square of the speed. With the speed remaining constant, the magnetic field is not changing the energy. This requires careful consideration when studying how magnets attract metals since the energy is changing in that setting.

## 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. Beam in a Cyclotron.

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.

## Key takeaways

• A charged moving particle is affected by a magnetic field. Both the charge and the movement are necessary for the field to exert a force.

• The force exerted by a magnetic field on a charged moving particle is known as Lorentz force. This is perpendicular to the direction of movement of the particle and to the magnetic field.

• A moving charged particle in a region with a uniform magnetic field describes a circular trajectory. However, its speed and energy remain unchanged.

• Cyclotrons and synchrotrons are particle accelerators based on the Lorentz force.

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

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

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.

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.

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.

## Moving Charges in a Magnetic Field Quiz - Teste dein Wissen

Question

Select the correct statement:

An electric charge generates an electric field. If it moves, a magnetic field appears, too.

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Question

Select the correct statement:

Electric charges are measured in Coulombs.

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Question

Select the correct statement:

The vector product is an operation between two vectors that yields a vector perpendicular to the other two.

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Question

Select the correct statement:

A moving charged particle in a uniform magnetic field describes a circular trajectory.

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Question

Select the correct statement:

Cyclotrons and synchrotrons use electric fields to linearly accelerate particles and a magnetic field to curve their trajectory.

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Question

What is the name of the force exerted by the electric and magnetic fields on a charge?

Lorentz force.

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Question

What is the name of the rule that helps to determine the direction of the vector obtained by a vector product?

Right-hand rule.

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Question

Is the order of the vector product irrelevant?

No, changing the order yields a global minus sign.

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Question

What requisites need to be imposed on a particle for it to be affected by a magnetic field?

It must have a charge, and it must be moving.

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Question

Is a moving charge affected by a magnetic field perpendicular to its velocity?

Yes, the Lorentz force is maximised in this case.

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Does a uniform magnetic field change the energy of a moving charge?

No, since it does not change its speed.

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Question

What movement does a moving charge describe when affected by a uniform magnetic field?

A circular trajectory.

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Question

How does one manage to initially accelerate a charge in a cyclotron?

With the help of an electric field.

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Question

How does one manage to periodically accelerate a charge in a cyclotron?

By increasing the value of the magnetic field.

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Question

Are there any relativistic effects in cyclotrons when approaching speeds close to the speed of light?

Yes, there are, and it is this that caused the development of synchrotrons.

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