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

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 previous lessons 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.

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.