[플라즈마 물리][Plasma Physics]CH2 - Single Particle Motion - Nonuniform B-field Curvature Drift

Consider a magnetic field that has curvature.

\(\bullet\) Such \(B\)-field will have gradient which result in the grad-B drift.

\(\bullet\) Once particle has a velocity along the magnetic field, it experiences a centrifugal force due to the field curvature and this force gives a drift.

The centrifugal force on a charged particle with \(\vec{v_{\parallel}}\) along \(\vec{B}\) is

\[\begin{aligned} \vec{F_{cf}}= m\frac{v_{\parallel}^2}{R_c}\hat{r} = \frac{ m v_{\parallel}^2}{R_c}\frac{\vec{R_c}}{R_c}\end{aligned}\]

where \(\vec{R_c}\) is the radius of curvature. The drift velocity due to \(\vec{F_{cf}}\) (the curvature drift) is then

\[\begin{aligned} \boxed{\vec{v_R} = \frac{mv_{\parallel}^2}{qB^2}\frac{\vec{R_c}\times \vec{B}}{R_c^2}}\end{aligned}\]

Curvature drift is \(\vec{v_R}\). Now, total drift in a curved magnetic field is the combination of the grad-B drift and the drift due to the centrifugal force. From Ampere’s Law, \(\nabla \times \vec{B} = \mu_0 \vec{J}\), we have \(\nabla \times \vec{B}=0\) in a vacuum. In the curvature, B-field is defined as \(\vec{B} = B(r) \hat{\phi}\)

\[\begin{aligned} \nabla \times \vec{B} &= \left[ \frac{1}{r} \frac{\partial B_z}{\partial \phi} - \frac{\partial B_{\phi}}{\partial z } \right]\hat{r} + \left[\frac{\partial B_r}{\partial z} - \frac{\partial B_z}{\partial r} \right] \hat{\phi} + \frac{1}{r}\left[ \frac{\partial}{\partial r}(r B_{\phi}) - \frac{\partial B_r}{\partial \phi} \right]\hat{z} \\ &= \frac{1}{r}\frac{d}{dr}(rB) =0\end{aligned}\]

Hence,

\[\begin{aligned} B(r) \propto \frac{1}{r} = \frac{1}{R_c}\end{aligned}\]

Thus,

\[\begin{aligned} \frac{\nabla B}{B} &= \frac{1}{B}\frac{\partial B}{\partial r} \hat{R_c} = -\frac{\vec{R_c}}{R_c^2}\end{aligned}\]

Then the grad-B drift velocity may be written as

\[\begin{aligned} \vec{\nabla B} = \frac{\frac{1}{2}mv_{\perp}^2}{qB} \frac{\vec{B} \times \nabla B}{B^2} = \frac{\frac{1}{2}mv_{\perp}^2}{qB}\frac{\vec{B}}{B} \times \left( \frac{-\vec{R_c}}{R_c^2} \right) = \frac{\frac{1}{2}mv_{\perp}^2}{qB^2}\frac{\vec{R_c}\times \vec{B}}{R_c^2}\end{aligned}\]

Finally, the total drift in a curved magnetic field is sum of grad-B drift and centrifugal force driven drift.

\[\begin{aligned} \boxed{\vec{v_R} + \vec{v_{\nabla B}} = \frac{m}{q}\frac{\vec{R_c}\times \vec{B}}{R_c^2 B^2} \left( v_{\parallel}^2 + \frac{1}{2}v_{\perp}^2 \right)}\end{aligned}\]

As you see, since the equation depends on the charge, we will have charge separation and leads to the rise of current. Notice that \(\frac{1}{2}v_{\perp}^2\) comes from the grad-B drift and \(v_{\parallel}^2\) comes from the centrifugal driven drift; the curvature drift.

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