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# Left / Right-Hand Rule: How to Determine Lorentz Force Direction

Using the right-hand rule, you can easily answer the following questions:

• In which direction does the Lorentz force \( \class{green}{\boldsymbol{F}} \) act on an electric charge moving in a magnetic field?

• In which direction does an electric current \( \class{red}{\boldsymbol{I}} \) flow?

• In which direction is the magnetic field \( \class{violet}{\boldsymbol{B}} \) directed?

• Was a positive or a negative charged particle deflected in the magnetic field? So you can also find out the charge type with the right-hand rule!

The right-hand rule is quite useful when working with magnetic fields and electric currents. Let's first clarify a few basics to reach the same level of knowledge.

## Ingredient #1: Moving charges

To apply the right-hand rule at all, you need moving electric charges. The emphasis is on the words "moving" and "charge". Don't you have any moving charges? No positively charged protons moving anywhere? No negatively charged electrons? No other positively or negatively charged particles? Then the right-hand rule is useless.

Usually, the movement of a charge, represented in a picture, is marked with a vector ("arrow"). This vector tells you in which direction the particle is moving. Usually next to the vector is its label, namely the letter \( \class{red}{\boldsymbol{v}} \). The letter is an abbreviation of the English word "\(\class{red}{\boldsymbol{v}}elocity\)". It is obvious, a moving charge, must have some velocity...

Sometimes you don't have just one moving charge, but many! These charges usually move in the same direction. This collective movement of electric charges in a particular direction, we call electric current.

In the lesson about electric current you have already learned what an electric current means. Electric current is abbreviated with the letter \(\class{red}{\boldsymbol{I}}\) and tells how many electric charges per second move through a cable, for example. The electric current thus describes the movement of many charged particles.

The current direction, i.e. the direction in which all charges move, is also represented by a vector (i.e. arrow). At the vector we write the letter \(\class{red}{\boldsymbol{I}}\) to say: "There is an electric current flowing here and in this direction".

You can apply the right-hand rule to one moving charge as well as to several moving charges, i.e. to the electric current. So it doesn't matter how many charges are moving, much more important is the question:

Do I even have a MOVING electric CHARGE or a CURRENT?

If you can answer this question with YES, then a necessary condition for the application of the right-hand rule is already fulfilled.

## Ingredient #2: Magnetic field

However, a moving charge alone is not enough. It would simply continue to move straight ahead. What you are still missing is the magnetic field. Abbreviated with the letter \( \class{violet}{\boldsymbol{B}} \). The magnetic field is like the velocity and the current a directed quantity. So we can ask ourselves not only in which direction the charge moves or in which direction the current flows, but also in which direction the magnetic field \( \class{violet}{\boldsymbol{B}} \) points. The magnetic field can also be assigned a vector ("arrow"), which indicates the direction of the magnetic field.

The direction of the magnetic field is defined to point from the north pole to the south pole. So if you take a horseshoe magnet, for example, you first have to find out where its north and south poles are. If you don't know that, then the right-hand rule can help you figure it out. You'll learn how to do that in a moment!

Sometimes you encounter two-dimensional images in which the magnetic field points into or out of the image.

• If you see a cross ⨂ (usually with a circle around it), this means a magnetic field pointing into the image (into the screen). The magnetic field points away from you.

• When you see point ⨀ with a circle around it, this means a magnetic field pointing out of the image (out of the screen). The magnetic field is directed towards you.

## Ingredient #3: Magnetic force (Lorentz force)

Now, if an electric charge moves orthogonally in a magnetic field, this charge is deflected. So it experiences a magnetic force \(\class{green}{\boldsymbol{F}}\) (Lorentz force) that acts on the charge, deflecting it. The exact direction in which the charge is deflected depends on whether it is a positive or a negative charge. A negative charge is deflected exactly opposite to a positive charge. The type of charge determines whether we use the left or the right hand rule.

### Do I use my left hand or my right hand?

If you look at your hands, you will see that they are mirrored. God did not provide us with mirrored hands by chance.

• The left hand he created for negative charges, like electrons.

• The right hand he created for positive charges, like protons.

Of course, this is also true for the current \(\boldsymbol{I}\): If, for example, positive charges travel through an electrical conductor, which is of course in the magnetic field, then you use the right hand. For the current of negative charges you use the left hand.

## How to apply left / right hand rule?

Important! You have to deal with three directions:

• With the direction of the moving charge or current \(\class{blue}{\boldsymbol{I}}\).

• With the direction of the magnetic field \(\class{violet}{\boldsymbol{B}}\).

• With the direction of deflection by Lorentz force \(\class{green}{\boldsymbol{F}}\).

TWO of the three directions should be given! You should also know whether you are dealing with positive or negative charges.

For example, you know the direction of \(\class{blue}{\boldsymbol{I}}\) and \(\class{violet}{\boldsymbol{B}}\) and you know that we are dealing with negatively charged electrons. Then you can find out the third direction, namely that of the Lorentz force \(\class{green}{\boldsymbol{F}}\), with the left / right hand rule. So far clear?

So you have one or more moving charges in the magnetic field. Good. Next, you should answer the following question:

Do I have to use the right or the left hand?

This question clarifies which type of charge (+ or -) you are dealing with. Once you have answered this, you only have to answer the very last question:

Which direction of the quantities \(\class{blue}{\boldsymbol{I}}\), \(\class{violet}{\boldsymbol{B}}\), \(\class{green}{\boldsymbol{F}}\) is unknown?

Use the fingers of your selected hand as follows:

• Thumb - stretch your thumb in the direction of the moving charge or in the direction of the current \(I\).

• Index finger - stretch your index finger, perpendicular to your thumb, in the direction of the magnetic field \( \class{violet}{\boldsymbol{B} } \). So in the direction of the magnetic south pole.

• Middle finger - extend your middle finger as perpendicular as possible to the other two fingers, then the middle finger will show you the direction of the Lorentz force \( \class{green}{\boldsymbol{F} } \)..

Why don't you try out the left / right hand rule on the following two illustrations...

• Quest #1: \(\class{blue}{\boldsymbol{v}}\) and \( \class{violet}{\boldsymbol{B} } \) directions are known. To which direction is the positive and negative charge deflected?

• Quest #2: This time \(\class{blue}{\boldsymbol{v}}\) and \( \class{green}{\boldsymbol{F} } \) directions are known. Where is the south pole of the magnet?

• Quest #3: Now pretend that you don't know whether the charge entered the magnetic field from the left or from the right. You only know in which direction the charge was deflected, the direction of \( \class{violet}{\boldsymbol{B} } \) as well as the charge type. From which direction must the charge have come?

• Quest #4: Now all three directions \(\class{blue}{\boldsymbol{v}}\), \( \class{violet}{\boldsymbol{B} } \) and \( \class{green}{\boldsymbol{F} } \) are known, but the charge type is not. Was a positively or a negatively charged particle deflected in the magnetic field?

## Example #1: Deflection of a current-carrying wire

You have a wire with current flowing through it with electrons. The wire is in a magnetic field which is directed into the screen.

Does the Lorentz force act on the wire to the left or to the right?

You have here an electron current. That means: You need your left hand.

1. Thumb - points in the direction of the electron flow.

2. Index finger - points into the screen.

3. Middle finger - then points to the right.

Thus, the Lorentz force acts on the wire to the right.

## Example #2: Deflection of the conductor swing

Take a horseshoe magnet. According to the definition, the magnetic field lines on the inside of the horseshoe magnet point from the north to the south pole. Then you take a wire, form it into a kind of circle or rectangle and put one side of the horseshoe magnet (in illustration 14 it is the side of the south pole) into the formed circle / rectangle. The wire should be deflectable inside the horseshoe magnet for the experiment to work at all. We call such a wire a conductor swing.

Will the conductor swing deflected inwards or outwards?

First of all, we deal here with a positive charge current. That means: You need your right hand.

1. Thumb - points in the direction of the positive current.

2. Index finger - points in the direction of the south pole of the horseshoe magnet.

3. Middle finger - then points to the outside of the horseshoe magnet.

The conductor swing thus experiences the Lorentz force to the outside.

In the next lesson, you will learn how exactly a moving charged particle is deflected by the Lorentz force in the magnetic field and why it causes the particle to move on a circular or spiral path.

## Exercises with Solutions

Use this formula eBook if you have problems with physics problems.

### Exercise #1: Proton in Magnetic Field

A proton moves into a magnetic field. Will it be deflected upwards or downwards?

#### Solution for Exercise #1

A proton moves to the right (velocity \( v \)) into the magnetic field (the magnetic field acts into the screen). The proton is deflected upwards by the Lorentz force.

### Exercise #2: Neutron in Magnetic Field

A neutron moves into a magnetic field. Where will the neutron be deflected?

#### Solution for Exercise #2

A neutron moves into a magnetic field. Where will the neutron be deflected?

Since the neutron is a neutral particle, it carries no electric charge (\( q = 0 \)). Therefore, no Lorentz force acts on the neutron, and it flies straight ahead - no matter how complex the magnetic field through which the neutron passes.

### Exercise #3: Wire Loop in Magnetic Field

You are moving a metal rod to the right. In which direction does the Lorentz force act on the electrons in the rod?

#### Solution for Exercise #3

You are moving the metal rod to the right. The metal rod is in a magnetic field that points into the screen, i.e., its direction is exactly perpendicular to the direction of the rod's displacement.

In the metal rod, there are free electrons which, as you know, are deflected by a magnetic force when they are moved. You move the metal rod to the right, so you also move the electrons sitting there to the right. Your task is to find out whether these electrons in the metal rod are deflected upwards or downwards along the metal rod.

If you want to find out the direction of deflection, you need to apply the three-finger rule. In this case, you use the left hand because it involves negatively charged electrons. Point your thumb (cause) in the direction of the displacement of the metal rod. Extend your index finger into the screen, i.e., in the direction of the magnetic field (effect). If you only stretch your middle finger perpendicular to the other fingers, it will point downwards. So you know - electrons are deflected downwards along the metal rod.

Since the metal rod is connected to wires and a lamp, the electrons move through the lamp and make it glow. Of course, only as long as you move the rod.

### Exercise #4: Current-Carrying Wire Swing in Magnetic Field

You insert a wire swing into a horseshoe magnet and let an electron current flow through it. Will the wire swing be displaced into or out of the horseshoe magnet?

#### Solution for Exercise #4

Electric current is caused here by the electrons. Therefore, we use the left hand. The wire swing is deflected into the horseshoe magnet.

### Exercise #5: Conductor Loop in Magnetic Field

You have a conductor loop divided into the upper half (where 2 electrons are depicted) and the lower half (where also 2 electrons are depicted).

You now rotate this conductor loop so that the upper part of the loop is moved into the screen, while the lower part of the loop is correspondingly moved out of the screen. The magnetic field runs from the north pole to the south pole, so downwards.

Determine the direction of the Lorentz force for each depicted electron. Does the rotation cause an electric current?

#### Solution for Exercise #5

In this problem, you use the left hand since it involves moving electrons.

First, consider the upper half of the conductor loop. According to the image, it is rotated into the screen. This means: An electron sitting in the upper part of the conductor loop will also be moved into the screen, so your thumb must reach into the screen.
The magnetic field is directed downwards, so your index finger of the left hand must point downwards.

The Lorentz force on the upper half of the conductor loop is thus - to the right!

As the upper half of the conductor loop is rotated into the screen, the lower half is correspondingly rotated out of the screen. Consequently, the lower 2 electrons are also moved out of the screen. The magnetic field still points downwards.

Thumb outwards, index finger downwards - yields a Lorentz force to the left!

The Lorentz force acts both on the upper and lower half along the conductor (except for the two lateral electrons, which, however, do not play a role in the current flow as they do not obstruct the current).

Therefore, the rotation of the conductor loop discussed above results in an electric current (electron current) clockwise.

### Exercise #6: Pendulum Ring in Magnetic Field

You want to pendulate a metallic ring - containing freely movable electrons - into a magnetic field. Assign a Lorentz force direction to each depicted electron!

#### Solution for Exercise #6

When swinging in, only half of the metallic ring is in the magnetic field. That means only the electron located in the magnetic field experiences a Lorentz force, which momentarily creates a clockwise current. Analogously, when swinging out. However, in the middle, both currents cancel each other out. Therefore, no current flows when the ring is completely in the magnetic field.