Learn how to determine the magnitude and direction of the magnetic force experienced by a charge moving in a magnetic field.

### Cross products

We'll need to understand cross products when working with magnetic forces. Like the dot product, the **cross product** is an operation between two vectors. Before getting to a formula for the cross product, let's talk about some of its properties.

We write the cross product between two vectors as

First, it is perpendicular to both

It's similar to the dot product, but instead of

There's one interpretation of the length of

### The right-hand rule

Notice that in the image above the cross product is perpendicular to

We have a convention called the **right-hand rule** to resolve this ambiguity. If you hold up your right hand, point your index finger in the direction of

It's arbitrary that we define the cross product with the right-hand rule instead of a left-hand rule, but by using this convention the cross product no longer has any ambiguity.

### Magnetic force on a moving charge

Now let's return to the topic of magnetic force on a charge. The magnetic force on a charge is described by the Lorentz force law, which is given by the vector cross product:

Yes. This is just the magnetic part of the Lorentz force law, which in its full form also includes the electric force:

Using the cross product relationship discussed earlier, we can write the *magnitude* of the magnetic force in terms of the angle

Notice that if the charged particle is not moving

The *direction* of the force can be found using the **right-hand rule** for the vector form of the equation.

*Note: If the moving charge is negative (for example, electrons) then you need to reverse the direction of your thumb because the force will be in the opposite direction.*

### Example: Magnetic deflection of electrons in a cathode-ray tube

A cathode ray tube is an evacuated tube with an electron gun at one end and a phosphorescent screen at the other end. Electrons are ejected from the electron gun at high speed and impact the screen where a spot of light is produced on impact with the phosphor.

Because electrons have charge, it is possible to deflect them in-flight with either the electric or magnetic force. Controlling the deflection allows the spot of light to be moved around the screen. Old style 'tube' televisions used this principle with magnetic deflection to form images by rapidly scanning the spot.

### Practice

The figure below shows a cathode ray tube experiment. A pair of coils are placed outside a cathode ray tube and produce a uniform magnetic field across the tube (not shown). In response to the field, the electrons are deflected and follow a path which is a segment of a circle as shown in the figure. What is the direction of the magnetic field?

The electrons emerge traveling to the *right.* This is the direction of *left.*

Additionally, we can see from the upward deflection of the electron beam that the electrons are experiencing a force *up.* This is the direction of

Applying the right-hand rule, the only way that *out of the page*.