[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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[플라즈마물리][Plasma Physics]CH2 - Single Particle Motion - Nonuniform B-field Grad-B drift

Case Five: Non Uniform Magnetic Field

Let us assume gradient B-field exist in the z-direction with gradient along the y-direction.

\[\begin{aligned} \vec{B}(r) = B(y)\hat{z}\end{aligned}\]

Assume \(B\) is slightly in-homogeneous.

\[r_L\left| \frac{\nabla B}{B} \right|<< 1\]

\(\vec{B}\) can be expressed by the Taylor series.

\[\begin{aligned} \vec{B} = \vec{B_0} + (\vec{r} \cdot \nabla)\vec{B} + ...\end{aligned}\]

so that we have

\[\begin{aligned} \vec{B} = \vec{B_0} + (\vec{r} \cdot \nabla)\vec{B} = \vec{B_0} + y\frac{\partial \vec{B}}{\partial y}\end{aligned}\]

From the Lorentz force, we will compute the average force and determine the guiding center drift.

\[\begin{aligned} \frac{dv_x}{dt} &=\frac{q}{m}v_y B(y) \hat{x}\\ \frac{dv_y}{dt} &=-\frac{q}{m}v_x B(y) \hat{y}\end{aligned}\]

\[\begin{aligned} v_x &= v_{\perp}\cos\omega_c t \\ v_y &= \mp v_{\perp}\sin\omega_c t\end{aligned}\]

\[\begin{aligned} x &= r_L \sin(\omega_c t) \\ y &=\pm r_L \cos \omega_c t\end{aligned}\]

Hence, using above equation, we can approximate the Lorentz force in inhomogeneous B-field.

\[\begin{aligned} F_x &= qv_y B =\mp q v_{\perp}\sin(\omega_c t) \left( B_0 \pm r_L \cos(\omega_c t)\frac{\partial B}{\partial y} \right)\\ F_y &=-qv_x B = -qv_{\perp}\cos\omega_c t \left(B_0 \pm r_L \cos(\omega_c t) \frac{\partial B}{\partial y} \right)\\\end{aligned}\]

Since

\[\begin{aligned} <\sin(\omega t)> &= \frac{1}{T} \int_{0}^{T}\sin \omega t dt = 0\\ <\cos(\omega t)> &= \frac{1}{T} \int_{0}^{T} \cos \omega t dt = 0\\ <\sin^2(\omega t)> &= \frac{1}{T} \int_{0}^{T}\sin^2 \omega t dt = \frac{1}{T} \int_{0}^{T} \frac{1-\cos(2\omega t)}{2}dt = \frac{1}{2}\\ <\cos^2(\omega t)> &= \frac{1}{T} \int_{0}^{T} \cos^2 \omega t dt = \frac{1}{T} \int_{0}^{T} \frac{1+\cos(2\omega t)}{2}dt = \frac{1}{2}\\ <\sin \omega t \cos \omega t> &=\frac{1}{T} \int_{0}^{T} \cos(\omega t) \sin(\omega t)dt = \frac{1}{T} \int_{0}^{T} \frac{1}{2}\sin(2\omega t)dt =0\end{aligned}\]

we get

\[\begin{aligned} <F_x> &=\frac{1}{T}\int_{0}^{T}F_x dt =0\\ <F_y> &=\frac{1}{T}\int_{0}^{T}F_y dt \\ &=\frac{1}{T}\int_{0}^{T}F_y dt \\ &=\mp qv_{\perp}r_L \frac{\partial B}{\partial y}\frac{1}{2}\\ &=-\mu \frac{\partial B}{\partial y}\end{aligned}\]

Notice that \(\mu = \pm \frac{1}{2}q v_{\perp} r_L\). By plugging this into the guiding center drift equation for general case,

\[\begin{aligned} \vec{v}_{\nabla B} &=\frac{\vec{F}\times \vec{B}}{qB^2} \\ &= \frac{F_y \hat{y} \times B \hat{z}}{qB^2}\\ &=\frac{F_y}{qB}\hat{x} \\ &= -\frac{\mu}{qB}\frac{\partial B}{\partial y}\hat{x}\end{aligned}\]

The expression can be generalized as

\[\begin{aligned} \vec{v}_{\nabla B} = \frac{\mu}{q}\frac{\vec{B} \times \nabla B}{B^2} \end{aligned}\]

or

\[\begin{aligned} \boxed{\vec{v}_{\nabla_B} = \frac{\frac{1}{2}mv_{\perp}^2}{qB}\frac{\vec{B} \times \nabla B}{B^2}} \end{aligned}\]

The grad-B drift depends of the charge \(q\), it can cause the plasma currents and charge separation.

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[플라즈마물리]Single Particle Motion - E cross B drift

Problem Consider a particle of charge \(q\) and mass \(m\), initially at rest at (0,0,0), in the presence of a static magntic field \(\vec{B}=B\hat{z}\) and \(\vec{E}=E\hat{y}\).

(a) Taking \(E, B > 0\), sketch the orbit of the particle when \(q>0\).

(b) Derive an exact expression for the orbit \([x(t),y(t),z(t)]\) or the particle. Express your answer in terms of \(E\), \(B\), and \(\omega_c\)

(c) Find the drift velocity after averaging the motion in time. If there were many particles of various charges and masses present, would there be any net current?

(d) Suppose the electric field were replaced by a force \(F\) in the \(y\)-direction. What would be the drift velocity?

Answer (a)

Answer (b)

\(\vec{B}=B\hat{z}\) and \(\vec{E}=E\hat{y}\). From Lorentz equation,

\[m\frac{dv}{dt}=q(\vec{E} + \vec{v}\times \vec{B})\]

\[\begin{aligned} \dot{v_x}&= \quad \quad \quad \quad \frac{q}{m}v_y B \nonumber \\ \dot{v_y}&= \frac{q}{m}\vec{E_y} - \frac{q}{m}v_xB \nonumber \\ \dot{v_z}&= 0\end{aligned}\]

In z-direction \[z(t) = constant = 0\]

\[\begin{aligned} \frac{dv_x}{dt} &= \frac{qB}{m}v_y = \omega_c v_y \nonumber \\ \frac{dv_y}{dt} &= \frac{qE}{m}-\frac{qB}{m}v_x = \frac{qE_y}{m}-\omega_c v_x \nonumber\end{aligned}\]

\[\begin{aligned} \frac{d^2 v_y}{dt^2} &= -\omega_c^2 v_y \nonumber \\ \frac{d^2 v_x}{dt^2} &= -\omega_c^2 \left(v_x-\frac{E}{B} \right) \nonumber\end{aligned}\]

since \(\frac{E}{B}\) is constant, \[\frac{d^2}{dt^2} \left[ v_x-\frac{E}{B} \right] = \frac{d^2 v_x}{dt^2} = -\omega_c^2 \left(v_x-\frac{E}{B} \right)\]

\[\begin{aligned} v_x &= iv_{\perp}e^{i\omega_c t} + \frac{E}{B}\nonumber \\ v_y &= v_{\perp}e^{i\omega_c t} \nonumber \end{aligned}\]

\[\begin{aligned} x(t) &= x(0) + r_L \cos{\omega_c t} +\frac{E}{B}t \nonumber \\ y(t) &= y(0) + r_L \sin{\omega_c t} \nonumber \end{aligned}\]

Answer (c)

\[<v_d> = \frac{\int_{-\infty}^{\infty}v_d f(v) dv}{\int_{-\infty}^{\infty}f(v) dv} = \frac{E}{B}\]

There will be no current because \(\vec{E} \times \vec{B}\) drifts the ions and electrons in the same direction.

Answer (d)

\[\begin{aligned} \vec{v_f}&= \frac{1}{q} \frac{\vec{F} \times \vec{B}}{B^2} \nonumber \\ &=\frac{FE}{qB}\hat{y}\end{aligned}\]

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