Radius at which a charged particle gyrates in a magnetic field.

The force **F** exerted on a charged particle of mass **m** and charge **q** and velocity **v** in a uniform magnetic field **B** is given by

**F**=md**v**/dt=q(**v**X**B**)

For simplicity, consider a magnetic field with a

**z** component only.

Acceleration in the

**z** direction is zero while acceleration in the x and y direction is given by

dv_{x}/dt=qBv_{y}/m=w_{c}v_{y}

dv_{y}/dt=qBv_{x}/m=w_{c}v_{x}

where w

_{c} is the

cyclotron frequency (eB/m). The solution to these

equations implies a Larmor Radius, L

L= √(2) mv_{thermal}/qB

where v

_{thermal} is the thermal velocity of the charged particles. Thus, the higher the magnetic field, the smaller the Larmor radius and the more tightly

bound the charged particle will be to the magnetic field. A

proton will gyrate at a larger Larmor radius than an

electron (its mass is much greater).

In order to contain a plasma, it is important to ensure that the device can contain particles gyrating at the Larmor radius.