## FANDOM

155 Pages

Curvature drift is a mode of motion of charged particles in a curved magnetic field. In contrast to the motion of charged particles in a uniform magnetic field, where the charged particles gyrate around the magnetic fields, in curvature drift the particles move mostly parallel to the magnetic field.

Consider a toroidal magnetic field of uniform intensity $\mathbf{B} = B \hat{\varphi}$. A charge particle in such a field can also move in a helical trajectory, with constant speed. The equations of motion are

$m \gamma \frac{d \mathbf{v}}{d t} = \frac{q}{c} \mathbf{v} \times \mathbf{B}$

The velocity is assumed to be of the form $\mathbf{v} = v_z \hat{z} + v_{\varphi} \hat{\varphi}$, where $v_z$ and $v_{\varphi}$ are constants. Substituting into the equation of motion, keeping in mind that $\frac{d \hat{\varphi} }{d t} = - \frac{v_{\varphi}}{r} \hat{r}$, where $r$ is the distance from the axis, yields

$v_z = \frac{\gamma m c}{q B r} v_{\varphi}^2$

In contrast to the helical motion in a uniform magnetic field, where the two components of the velocity are independent.