Gyro motion

From QED

< PlasmaWiki(Link to this page as [[PlasmaWiki/Gyro motion]])
Jump to: navigation, search

When charged particles have velocities perpendicular to a magentic field, they undergo a circular gyro motion, whose direction may be determined by the right-hand rule.

Let us suppose we have a uniform magnetic field \mathbf{B}=B\hat{z}. The equation of motion will come from the Lorentz force :

m\dot{\mathbf{v}}=\frac{q}{c} \mathbf{v}\times\mathbf{B}

For all three directions:

\begin{matrix} m\dot{v}_x = \frac{q}{c} v_y B \\ m\dot{v}_y = -\frac{q}{c} v_x B \\ m\dot{v}_z = ; 0 \end{matrix}

We can substitute in to get equations in just one variable:

\begin{matrix} \ddot{v}_x = -\left(\frac{qB}{m c}\right)^2 v_x \\ \ddot{v}_y = -\left(\frac{qB}{m c}\right)^2 v_y \end{matrix}

These equations have as their solution:

v_\perp = A\cos(\Omega_c t) + B\sin(\Omega_c t) + C

Where Ωc is the cyclotron frequency for the particle:

\Omega_c=\frac{qB}{mc}

This page was recovered in October 2009 from the Plasmagicians page on Gyro_motion dated 01:25, 19 October 2006.

Retrieved from "https://qed.princeton.edu/main/PlasmaWiki/Gyro_motion"
Personal tools