# PHY C22: Charged Particles in Magnetic Fields

So far we’ve explored forces on permanent magnets & current-carrying conductors. But what about free-moving charged particles?

• Force on a charge moving in a magnetic field
• F = Bqv sin θ
• Determination of v & e/me for electrons

First, let’s introduce ourselves to a few charged particles:

• electron
• proton
• α-particle (Helium-4 nucleus)
• β-particle (fast-moving nuclear electron)

These particles exhibit interesting effects when they pass through a magnetic field.

What is the force on a moving charged particle in a magnetic field?

## F = Bqv sin θ

Here’s a derivation:

Remember the motor effect:
F = BIL sin θ

Current I is given by the number of particles (n) of charge q passing through a point in time t:
I = nq/t

Subbing this in:
F = (Bnq/t)L sin θ

Since v = L/t,
F = Bnqv sin θ

If we are looking at the force acting on a single particle, n = 1, so
F = Bqv sin θ

For a charged particle moving perpendicular to a magnetic field,
F = Bqv

How does this effect the path of the particle?
It causes deflection.

The particle will travel across an arc of a circle. This is an example of Circular Motion (read up on it here).

We can calculate the radius of this arc as follows:

Here, the centripetal force is provided by the magnetic force:
Fc = FB
mv2/r = Bqv

Rearranging this,

The exact shape of the particle’s path depends on many factors.
Here are a few:

One important application of this is learning about the properties of newly-discovered charged particles.

Let’s take the electron as an example.
The mass-charge ratio of an electron is known as its specific charge.

It is given by e/me

To do this calculation, we must use a fine-beam tube:

• electrons are accelerated from rest using a high voltage V
• Helmholtz coils provide a permanent magnetic field B perpendicular to the beam
• the circular orbit of the electron beam can be seen as electrons strike low-pressure gas inside the tube, making it glow

Calculating the mass-charge ratio of a charged particle (electron)

A great simulation if you would like to play with the values of b, m, q, & v to see how they effect a particle’s motion in a magnetic field: oPhysics

In the next post, we will look at what happens when a charged particle enters a magnetic field AND an electric field.