What charge does a magnetic field generate

Lorentz force: how charge is deflected in a magnetic field

Lorentz force is generally the sum of electrical and magnetic forcethat hit an electrically charged particle with the charge \ (q \) works when dealing with the speed \ (\ class {blue} {v} \) in one Magnetic field \ (\ class {violet} {B} \) and in one electric field \ (E \) moves. Depending on how the direction of movement of the particle is relative to the direction of the magnetic field, the Lorentz force is different. This is made possible by the angle \ (\ alpha \) between \ (\ class {blue} {v} \) and \ (\ class {violet} {B} \) determined.

Amount of the Lorentz force (general)

Electrical component of the Lorentz force

The first term in the formula for Lorentz force stands for electrical force \ (F _ {\ text e} \):

Formula: electric force
Example: electron and protonIf you bring a positively charged proton close to a negatively charged electron, the electron experiences an electrical force \ (F _ {\ text e} \) and moves towards the proton. The same thing happens with the proton: It also experiences a force that the electron exerts on it. The two generate inhomogeneous (i.e. location-dependent) electric fields. In this way the charges influence each other electrically.

The electric field strength \ (E \) states what force a charge can exert on another charge, i.e.: Force per charge:

The field is strongest close to the charge and becomes weaker the further the electrical charges move away from each other. At the location of the electron the electric field generated by the proton has a certain value \ (F _ {\ text e} \).

If there is no external electric field \ (E \) in which the charge moves, then the electric part of the Lorentz force vanishes: \ (F _ {\ text e} = q \ cdot0 = 0 \). Then only the magnetic part \ (F _ {\ text m} \) remains.

Magnetic part of the Lorentz force

The second summand in the Lorentz force stands for the magnetic force \ (\ class {green} {F _ {\ text m}} \), which acts on an electrical charge \ (q \) in the magnetic field \ (\ class {violet} {B} \) when the charge interacts with the speed \ (\ class {blue} {v} \) moves:

Magnetic force

In the following we assume that the charge is only in a magnetic field \ (\ class {violet} {B} \). That means, there is no electric field \ (E \) and therefore no electric force on the charge. Then Lorentz force \ (F \) is equal to the magnetic force \ (\ class {green} {F _ {\ text m}} \):

In order for a magnetic force \ (\ class {green} {F} \) to act on a particle, it must meet the following properties:

  1. The particle has to move - otherwise the velocity would be \ (\ class {blue} {v} ~ = ~ 0 \) and thus also the Lorentz force:
  2. The particle must not be neutral - because neutral particles have no charge \ (q ~ = ~ 0 \). Therefore the Lorentz force would also disappear in this case:

When you have made sure that the particle fulfills the above two properties, then you can calculate the magnitude of the Lorentz force (in). In principle 3 cases can occur. The load is moving ...

  1. parallel to the magnetic field: \ (\ class {blue} {v} \) || \ (\ class {violet} {B} \)
  2. perpendicular to the magnetic field: \ (\ class {blue} {v} \) ⊥ \ (\ class {violet} {B} \)
  3. oblique to the magnetic field: at the angle \ (\ alpha \)

Case 1: Movement parallel to the magnetic field

In this case the angle that is in the formula for Lorentz force is: \ (\ alpha \) = 0. A sine of 0 degrees is 0, which is why there is no magnetic force on the particle and it therefore disappears:

Case 2: Movement perpendicular to the magnetic field

If two vectors (such arrows and so on) - in this case speed \ (\ class {blue} {v} \) and magnetic flux density \ (\ class {violet} {B} \) - are perpendicular to one another, then that means that they enclose a \ (90 ^ \ circ \) angle. And as you may know from mathematics: A sine of 90 degrees is 1. Therefore you can write in a simplified way:

Formula: Lorentz force - movement perpendicular to the magnetic field

If you also have the direction of the force - without vector calculation - then use the three-finger rule!

You can only use the three-finger rule with which you can determine the direction of the Lorentz force in case 2! In two other cases it does not apply! Brief repetition of the three-finger rule:

  • thumb - points in the direction of the cause, here movement of the charge, i.e. in the direction of the velocity \ (\ class {blue} {v} \).
  • index finger - points in the direction of the magnetic south pole (mostly marked with green in school).
  • Middle finger - shows you the Lorentz force direction as soon as you have correctly directed the other two fingers.

For positive charges (e.g. protons) you have to use your right hand and for negative charges (e.g. electron) you have to use the left hand.

Creation of circular motion

For example, if you use an electron gun to shoot a negative charge \ (q = -e \) into a magnetic field \ (\ class {violet} {B} \) directed into your screen, in such a way that the charge with a constant velocity \ (\ class {blue} {v} \) vertically (ie case 2) enters the magnetic field, then it experiences - as you know - a Lorentz force and is deflected upwards. However, the load does not just fly straight up, but runs through a circular path, because:

Calculate the radius of the circular path

If you want to calculate the radius \ (r \) of the circular path, you simply equate the Lorentz force with the centripetal force: Like the centripetal force, the Lorentz force always acts in the center of the circle, which is why it replaces the centripetal force here. Form the equation according to \ (r \), then you have:

Formula: radius of the circular path

The formula gives you some useful information about the circular path radius.

You can judge the strength of the magnetic field, for example, by looking at the radius of the circular path that has arisen. Because the larger the radius, the weaker the magnetic field.

Period of the circular motion

If you still want to calculate the period \ (T \), i.e. the time that the particle needs to make exactly one circular motion, then you use the formula for uniform motion:

This formula is allowed here, since the amount of the speed of the particle (but not its direction!) Is constant at any point in time, which is why it is actually a uniform and not an accelerated movement.

The segment \ (s \) - is the circumference of the circle, so: \ (s = 2 \ pi \, r \). Time \ (t \) is the period \ (T \). The period indicates how long a cycle lasts.

You now have everything you need! Plug in the circumference \ (s \) in. The time \ (t \) in the searched period is \ (T \): \ (t = T \). Also plug in the radius in:

Formula: Period of a circular movement

Case 3: Movement at an angle to the magnetic field

You are interested in the magnitude of the magnetic force on a charge that does not necessarily move exactly perpendicular to the magnetic field. The charge could somehow move partially parallel to the magnetic field. Therefore you consider:

Formula: Amount of the Lorentz force - any entry angle

If the speed \ (\ class {blue} {v} \) is directed obliquely to the magnetic flux density \ (\ class {violet} {B} \), then speed can be in a parallel \ (\ class {blue} {v_ {||}} \) and one vertical \ (\ class {blue} {v _ {\ perp}} \) part of the magnetic field.

The parallel part - in contrast to the vertical part - has no influence on the magnetic force and therefore this part is not responsible for the deflection of the electron in the magnetic field; because the vertical part forms an angle of 0 degrees with the magnetic flux density \ (\ class {violet} {B} \), which is why the force for this part disappears (because of \ (\ sin (0 ^ {\ circ}) ~ = ~ 0 \)):

A partial movement parallel and a partial movement perpendicular to the magnetic field creates a cylindrical spiral path, a so-called Helix. Its axis is parallel to the magnetic field. It has a radius \ (r \) and a pitch \ (h \). Where the pitch is simply a distance parallel to the magnetic field that is covered within a period \ (T \).

Lorentz force on a current-carrying conductor

Lorentz force acts not only on individual charges, but also on entire electrical currents! They represent nothing more than electrical charges that move, for example, through an electrical conductor.

Charges in the conductor (which represent the electric current) cover the length of the conductor \ (L \) within a certain time \ (t \). Distance per time is defined as speed (in this case the speed of the charges in the conductor):

Inserting the speed into the Lorentz force formula and rearranging \ (t \) results in: And \ (\ frac {q} {t} \) is defined as the current strength \ (\ class {blue} {I} \). In total you have:

Lorentz force on a current-carrying conductor

Lorentz force between two ladders

Imagine two electrical cables parallel to each other and let the electrical current \ (\ class {blue} {I_1} \) flow through one and \ (\ class {blue} {I_2} \) in the same direction through the other. You will find that the two conductors tighten due to the Lorentz force. But how - without a magnetic field? The reason is:

The magnetic field generated by the charges in the conductor encompasses the conductor. In addition, the generated magnetic field is not concentrated in a specific place, but rather extensive. Because of this expansion, the other conductor is suddenly in an external magnetic field.

If you proceed in the same way with the other conductor, you will find that the magnetic field at the location of the other conductor points in the opposite direction, so that the Lorentz force also points in the opposite direction than with the other conductor.

What happens if the electrical currents in the conductors go in opposite directions?

You can calculate the force that each conductor experiences - due to the other conductor - as follows:

Formula: Lorentz force between two conductors