Magnetic force on moving charges

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 $\overrightarrow{a}\times \overrightarrow{b}$‍ (pronounced "a cross b"). Unlike the dot product, which returns a number, the result of a cross product is another vector. Let's say that $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$‍. This new vector $\overrightarrow{c}$‍ has a two special properties.

First, it is perpendicular to both $\overrightarrow{a}$‍ and $\overrightarrow{b}$‍. Second, the length of $\overrightarrow{c}$‍ is a measure of how far apart $\overrightarrow{a}$‍ and $\overrightarrow{b}$‍ are pointing, multiplied by their magnitudes.

It's similar to the dot product, but instead of $\cos (\theta )$‍ the cross product uses $\sin (\theta )$‍, where $\theta$‍ is the angle between $\overrightarrow{a}$‍ and $\overrightarrow{b}$‍. That way, when the angle is $90$‍ degrees, the cross product is at its largest. In this sense, the dot product and the cross product complement each other.

There's one interpretation of the length of $\overrightarrow{c}$‍ that is particularly useful. Think of the parallelogram formed by $\overrightarrow{a}$‍ and $\overrightarrow{b}$‍. The base of this parallelogram has length $\Vert \overrightarrow{a}\Vert$‍, and the height has length $\Vert \overrightarrow{b}\Vert \sin (\theta )$‍. That means the area of the parallelogram in total is precisely the magnitude of the cross product.

Notice that in the image above the cross product is perpendicular to $\overrightarrow{a}$‍ and $\overrightarrow{b}$‍, as expected. But there are actually two vectors that could be perpendicular to $\overrightarrow{a}$‍ and $\overrightarrow{b}$‍. If $\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}$‍, then these two choices are $\overrightarrow{c}$‍ and $-\overrightarrow{c}$‍. How do we decide which of the two perfectly valid choices is the cross product?

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 $\overrightarrow{a}$‍, and point your middle finger in the direction of $\overrightarrow{b}$‍, then your thumb will point in the direction of $\overrightarrow{a}\times \overrightarrow{b}$‍.

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.

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:

Using the cross product relationship discussed earlier, we can write the magnitude of the magnetic force in terms of the angle $\theta$‍ ($<{180}^{\circ }$‍) between the velocity vector and the magnetic field vector:

Notice that if the charged particle is not moving $(v=0),$‍ then the magnitude of the magnetic force it experiences is zero.

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

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.

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?