From ea14007fd87b5ffc833ca820347842e4886a75bc Mon Sep 17 00:00:00 2001 From: Krishna Ayyalasomayajula Date: Sat, 2 May 2026 12:52:07 -0500 Subject: [PATCH] =?UTF-8?q?fix(HJ):=20A.12=20=E2=80=94=20Explain=20gauge?= =?UTF-8?q?=20choice,=20guiding=20center=20physics,=20averaging,=20univers?= =?UTF-8?q?ality?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- concepts/advanced/crossed-fields-hj.tex | 19 +++++++++++++++---- 1 file changed, 15 insertions(+), 4 deletions(-) diff --git a/concepts/advanced/crossed-fields-hj.tex b/concepts/advanced/crossed-fields-hj.tex index 37c00a2..7b087ef 100644 --- a/concepts/advanced/crossed-fields-hj.tex +++ b/concepts/advanced/crossed-fields-hj.tex @@ -21,7 +21,7 @@ becomes explicitly \] The coordinates $x$ and $z$ do not appear in $\mcH$, so they are cyclic and their conjugate momenta $p_x$ and $p_z$ are conserved.} -\nt{Choice of gauge does not affect physical observables. The Landau gauge $\vec{A} = (-B_0 y, 0, 0)$ breaks translational symmetry in $y$ but preserves it in $x$, making $p_x$ the conserved momentum. An alternative symmetric gauge $\vec{A} = \tfrac12 B_0(-y, x, 0)$ would be natural for purely rotational problems but obscures the drift structure we want to expose here.} +\nt{Gauge choice for crossed fields. The Landau gauge $\vec{A} = (-B_0 y, 0, 0)$ differs from the gauge used in the pure magnetic-field problem (subsection A.11), where cylindrical symmetry guided the choice. Here, placing the electric field along the $y$-direction makes this particular Landau gauge algebraically convenient: because $\vec{A}$ depends only on $y$, the coordinate $x$ remains cyclic and $p_x$ is automatically conserved. This conserved momentum $\alpha_x$ couples directly into the guiding-centre position, allowing the drift velocity to emerge cleanly from the Hamilton--Jacobi formalism without solving coupled differential equations.} \thm{$E \times B$ drift velocity from the guiding centre}{ For crossed fields $\vec{E} = E_0\,\hat{\bm{y}}$ and $\vec{B} = B_0\,\hat{\bm{z}}$, the guiding centre of the orbit lies at @@ -76,6 +76,7 @@ The shift in brackets is the guiding-centre coordinate: \[ y_c = \frac{mE_0/B_0 - \alpha_x}{qB_0}. \] +Completing the square has a clear physical meaning: it reveals the equilibrium position $y_c$ at which the electric force $qE_0\,\hat{\bm{y}}$ is exactly balanced by the magnetic force arising from the guiding-centre's $x$-velocity. At $y = y_c$, the canonical momentum combination $\alpha_x + qB_0 y_c = mE_0/B_0$ gives precisely the velocity needed for the magnetic Lorentz force to cancel the electric force. Deviations from $y_c$ are therefore harmonic oscillations about this equilibrium. Substituting back, the full equation becomes \[ \bigl(\der{W_y}{y}\bigr)^2 + q^2B_0^2\bigl(y - y_c\bigr)^2 = 2mE - \alpha_z^2 - \alpha_x^2 + q^2B_0^2 y_c^2. @@ -88,14 +89,24 @@ This is precisely the Hamilton--Jacobi equation for a harmonic oscillator in the \[ \omega_c = \frac{|q|B_0}{m}. \] -The $y$-motion oscillates sinusoidally about $y_c$ with the cyclotron frequency. The time average of the position is the guiding centre, $\langle y \rangle = y_c$. +The $y$-motion oscillates sinusoidally about $y_c$ with the cyclotron frequency. The explicit time dependence of the coordinate follows from inverting the generating function: +\[ +y(t) = y_c + A\sin\bigl(\omega_c t + \varphi\bigr), +\] +where $A$ is the oscillation amplitude determined by the total energy and $\varphi$ is a phase set by initial conditions. Trigonometric averaging over one cyclotron period $T_c = 2\pi/\omega_c$ gives +\[ +\langle y \rangle = \frac{1}{T_c}\int_0^{T_c} y(t)\,\dd t += y_c + \frac{A}{T_c}\int_0^{T_c} \sin\bigl(\omega_c t + \varphi\bigr)\,\dd t += y_c, +\] +because the sine function integrates to zero over a complete period. The guiding-centre position is thus the exact time-average of the $y$-coordinate. Now compute the kinematic $x$-velocity. From Hamilton's equation, $\dot{x} = \pdv{\mcH}{p_x}$: \[ v_x = \pdv{\mcH}{p_x} = \frac{1}{m}\bigl(p_x + qB_0 y\bigr). \] -The canonical momentum $p_x = \pdv{\mcS}{x} = \alpha_x$ is constant. Averaging over one cyclotron orbit, $\langle y \rangle = y_c$, so the drift velocity is +The canonical momentum $p_x = \pdv{\mcS}{x} = \alpha_x$ is constant. With the trigonometric average $\langle y \rangle = y_c$ established, the drift velocity follows from substituting $y_c$ into the expression for $v_x$: \[ \langle v_x \rangle = \frac{\alpha_x + qB_0 y_c}{m}. \] @@ -140,7 +151,7 @@ With $\vec{B} = B_0\,\hat{\bm{z}}$ and $\vec{E} = E_0\,\hat{\bm{y}}$: \] This matches the Hamilton--Jacobi result exactly. The cross product $\vec{E}\times\vec{B}$ determines the drift direction and division by $B^2$ converts the magnitude into a velocity. Both formalisms predict the same drift regardless of the particle's charge or mass.} -\nt{A charge-sign reversal changes both the sense of Larmor rotation and the location of the guiding centre $y_c$, but these two effects exactly compensate in the averaged $x$-velocity. An electron and a proton spiralling in the same crossed fields therefore share the same guiding-centre drift, even though their individual gyroradii and rotation frequencies differ enormously. This universality is what makes the $E\times B$ drift so important in plasma physics: bulk plasma drifts as a coherent fluid.} +\nt{Universality of the $E \times B$ drift velocity. The drift velocity $v_d = E_0/B_0$ is independent of both the particle's mass and its charge sign. A charge-sign reversal changes the sense of Larmor rotation and shifts the guiding centre $y_c$, but these two effects exactly compensate in the time-averaged $x$-velocity. An electron and a proton spiralling in the same crossed fields share the same guiding-centre drift, even though their gyroradii and cyclotron frequencies differ enormously. This universality is precisely why the result is called the $E\times B$ drift velocity: the expression $\vec{v}_d = \vec{E}\times\vec{B}/B^2$ contains no mass or charge, so every charged species drifts together. In plasma physics this means bulk plasma moves as a coherent fluid rather than separating into counter-streaming components.} \mprop{Properties of the $E \times B$ drift}{ The drift velocity $\vec{v}_d = \vec{E}\times\vec{B}/B^2$ satisfies the following properties: