Single-Particle Motion in Uniform Fields & Electric Drifts

1. The Lorentz Equation of Motion

The classical trajectory of a point particle of rest mass $m$ and electric charge $q$ moving with velocity $\vec{v}$ in macroscopic electric $\vec{E}$ and magnetic $\vec{B}$ fields is governed by Newton's second law with the Lorentz force:

$$m \frac{d\vec{v}}{dt} = q\left( \vec{E} + \vec{v}\times\vec{B} \right)$$

Taking the scalar dot product with the velocity $\vec{v}$:

$$m \vec{v}\cdot\frac{d\vec{v}}{dt} = \frac{d}{dt}\left( \frac{1}{2}m v^2 \right) = q \vec{v}\cdot\vec{E} + q \vec{v}\cdot(\vec{v}\times\vec{B}) = q \vec{v}\cdot\vec{E}$$

Because $\vec{v}\cdot(\vec{v}\times\vec{B}) \equiv 0$, a pure magnetic field does zero work on a charged particle; it alters only the direction of the velocity vector while conserving total kinetic energy $\frac{1}{2}m v^2 = \text{const}$.

2. Gyration in a Static Uniform Magnetic Field

Let $\vec{B} = B_0 \hat{z}$ be static and spatially uniform, with $\vec{E} = 0$. Resolving the equation of motion into Cartesian components:

$$m \frac{dv_x}{dt} = q B_0 v_y, \quad m \frac{dv_y}{dt} = -q B_0 v_x, \quad m \frac{dv_z}{dt} = 0$$

Along the magnetic field, the parallel velocity is constant:

$$v_z(t) = v_\parallel = \text{const} \implies z(t) = z_0 + v_\parallel t$$

Differentiating the $x$-component and substituting $dv_y/dt$:

$$\frac{d^2 v_x}{dt^2} = \frac{q B_0}{m} \frac{dv_y}{dt} = \frac{q B_0}{m} \left( -\frac{q B_0}{m} v_x \right) = -\left( \frac{q B_0}{m} \right)^2 v_x$$

Defining the cyclotron frequency (or gyrofrequency) $\omega_c$:

$$\omega_c \equiv \frac{|q| B_0}{m}$$

The transverse velocity components oscillate harmonically at $\omega_c$:

$$\frac{d^2 v_x}{dt^2} + \omega_c^2 v_x = 0, \quad \frac{d^2 v_y}{dt^2} + \omega_c^2 v_y = 0$$

1. Larmor Radius & Gyration Circle

Choosing initial conditions such that $v_x(0) = v_\perp \cos\delta$ and integrating:

$$v_x(t) = v_\perp \cos(\mp \omega_c t + \delta), \quad v_y(t) = \pm v_\perp \sin(\mp \omega_c t + \delta)$$

where the upper sign corresponds to positive ions ($q > 0$) and the lower sign to electrons ($q = -e < 0$). Integrating the velocity components yields the spatial coordinates:

$$x(t) = X_0 + \frac{v_\perp}{\omega_c} \sin(\mp \omega_c t + \delta) = X_0 + r_L \sin(\mp \omega_c t + \delta)$$ $$y(t) = Y_0 \mp \frac{v_\perp}{\omega_c} \cos(\mp \omega_c t + \delta) = Y_0 \mp r_L \cos(\mp \omega_c t + \delta)$$

where $(X_0, Y_0)$ represents the fixed center of gyration, known as the guiding center. The radius of the gyration circle is the Larmor radius (or gyroradius) $r_L$:

$$r_L \equiv \frac{v_\perp}{\omega_c} = \frac{m v_\perp}{|q| B_0}$$

2. Helicity and Sense of Gyration

Looking along the direction of $\vec{B}$ (the $+z$ axis):

• Positive Ions ($q > 0$): Gyrate in a counter-clockwise sense (left-handed rotation).
• Electrons ($q < 0$): Gyrate in a clockwise sense (right-handed rotation).

Because an orbiting charge forms an infinitesimal circular current loop $I = q (\omega_c / 2\pi)$, the magnetic dipole moment $\vec{\mu} = I \vec{A}$ produced by the orbiting particle is directed opposite to the background magnetic field $\vec{B}$ for both ions and electrons:

$$\vec{\mu} = -\frac{m v_\perp^2}{2 B^2} \vec{B}$$

Consequently, a plasma of gyrating particles is fundamentally diamagnetic: particle gyration naturally generates an opposing internal magnetic field that reduces the ambient $\vec{B}$.

1. Coupled Equations of Motion

Now introduce a static, uniform electric field perpendicular to $\vec{B}$. Let $\vec{B} = B_0 \hat{z}$ and $\vec{E} = E_y \hat{y}$. The Lorentz equations of motion become:

$$m \frac{dv_x}{dt} = q B_0 v_y, \quad m \frac{dv_y}{dt} = q E_y - q B_0 v_x, \quad m \frac{dv_z}{dt} = 0$$

Differentiating the $x$-equation with respect to time:

$$\frac{d^2 v_x}{dt^2} = \frac{q B_0}{m}\frac{dv_y}{dt} = \frac{q B_0}{m}\left[ \frac{q E_y}{m} - \frac{q B_0}{m}v_x \right] = -\omega_c^2 \left( v_x - \frac{E_y}{B_0} \right)$$

Defining a shifted velocity coordinate $u_x \equiv v_x - \frac{E_y}{B_0}$:

$$\frac{d^2 u_x}{dt^2} + \omega_c^2 u_x = 0$$

This demonstrates that in the moving reference frame, the particle undergoes ordinary cyclotron gyration around a guiding center translating steadily along the $+x$-axis with constant velocity:

$$v_E = \frac{E_y}{B_0}$$

1. General Vector Derivation of E x B Drift

To derive the guiding center drift velocity in general coordinate-free vector form, partition the particle velocity $\vec{v}$ into a slowly varying guiding center drift velocity $\vec{v}_E$ and a rapidly oscillating gyration velocity $\vec{v}_c$:

$$\vec{v} = \vec{v}_E + \vec{v}_c$$

Averaging the Lorentz force equation over one complete cyclotron gyration period $\tau_c = 2\pi / \omega_c$, the periodic gyration acceleration averages to zero: $\langle d\vec{v}_c / dt \rangle = 0$. For a steady drift ($d\vec{v}_E / dt = 0$):

$$0 = q \left( \vec{E} + \vec{v}_E \times \vec{B} \right)$$

Taking the vector cross product with $\vec{B}$ on both sides:

$$\vec{E} \times \vec{B} + (\vec{v}_E \times \vec{B}) \times \vec{B} = 0$$

Applying the vector triple product identity $(\vec{A}\times\vec{B})\times\vec{C} = (\vec{A}\cdot\vec{C})\vec{B} - (\vec{B}\cdot\vec{C})\vec{A}$:

$$(\vec{v}_E \times \vec{B}) \times \vec{B} = (\vec{v}_E \cdot \vec{B})\vec{B} - B^2 \vec{v}_E = -B^2 \vec{v}_{E,\perp}$$

Assuming the drift is purely perpendicular to $\vec{B}$ ($\vec{v}_E \cdot \vec{B} = 0$):

$$\vec{E} \times \vec{B} - B^2 \vec{v}_E = 0 \implies \vec{v}_E = \frac{\vec{E} \times \vec{B}}{B^2}$$

2. Fundamental Physical Properties of E x B Drift

The $\vec{E}\times\vec{B}$ drift possesses several remarkable physical properties:

1. Charge Independence: The drift velocity $\vec{v}_E$ is completely independent of the sign of the electric charge $q$. Both positive ions and negative electrons drift in the identical direction with the identical velocity.
2. Mass Independence: The drift velocity does not depend on the particle mass $m$. Protons, heavy impurities, and electrons all drift together.
3. Absence of Net Electric Current: Because both species move in unison: $$\vec{J}_E = n_e q_e \vec{v}_{E,e} + n_i q_i \vec{v}_{E,i} = n_0 (-e) \vec{v}_E + n_0 (+e) \vec{v}_E = 0$$ The $\vec{E}\times\vec{B}$ drift produces zero net electric current in a quasi-neutral plasma.

1. Generalized Guiding Center Force Drift

Any general non-electromagnetic force $\vec{F}$ (such as gravity $\vec{F}_g = m\vec{g}$ or centrifugal force $\vec{F}_c = m v_\parallel^2 \hat{R}_c / R_c$) acting on a charged particle can be represented by an effective electric field:

$$\vec{E}_{\text{eff}} = \frac{\vec{F}}{q}$$

Substituting $\vec{E}_{\text{eff}}$ into the drift formula yields the generalized force drift velocity $\vec{v}_F$:

$$\vec{v}_F = \frac{\vec{E}_{\text{eff}} \times \vec{B}}{B^2} = \frac{1}{q} \frac{\vec{F} \times \vec{B}}{B^2}$$

2. Gravitational Drift

For a constant gravitational acceleration $\vec{g}$:

$$\vec{v}_g = \frac{m}{q} \frac{\vec{g} \times \vec{B}}{B^2}$$

Crucially, because of the explicit factor of $q$ in the denominator:

• Ions and electrons drift in opposite directions!
• Because $m_i \gg m_e$, the ion gravitational drift velocity is larger than the electron drift velocity by the mass ratio $M_i / m_e \approx 1836$.
• This opposite motion establishes a net macroscopic cross-field electric current density: $$\vec{J}_g = n_0 e (\vec{v}_{gi} - \vec{v}_{ge}) \approx n_0 (M_i + m_e) \frac{\vec{g}\times\vec{B}}{B^2} \approx \rho_m \frac{\vec{g}\times\vec{B}}{B^2}$$ where $\rho_m = n_0 M_i$ is the mass density of the plasma.

1. Inertial Lag in Time-Varying Fields

Consider a slowly time-varying electric field $\vec{E}(t) \perp \vec{B}_0$, where the rate of change is much slower than the cyclotron frequency:

$$\omega \ll \omega_c \iff \left| \frac{1}{E}\frac{dE}{dt} \right| \ll \omega_c$$

As $\vec{E}$ changes, the guiding center $\vec{E}\times\vec{B}$ drift velocity $\vec{v}_E(t) = \frac{\vec{E}(t)\times\vec{B}}{B^2}$ must also accelerate with time:

$$\frac{d\vec{v}_E}{dt} = \frac{1}{B^2}\left( \frac{d\vec{E}}{dt} \times \vec{B} \right)$$

Because the particle possesses finite mass $m$, inertia resists this acceleration. The particle experiences an effective inertial D'Alembert force:

$$\vec{F}_{\text{inertial}} = -m \frac{d\vec{v}_E}{dt} = -\frac{m}{B^2}\left( \frac{d\vec{E}}{dt} \times \vec{B} \right)$$

2. The Polarization Drift Velocity

This inertial force drives an additional guiding center drift according to the general force drift formula:

$$\vec{v}_p = \frac{1}{q}\frac{\vec{F}_{\text{inertial}} \times \vec{B}}{B^2} = -\frac{m}{q B^4}\left[ \left( \frac{d\vec{E}}{dt} \times \vec{B} \right) \times \vec{B} \right]$$

Applying the vector triple product identity $(\vec{A}\times\vec{B})\times\vec{B} = -B^2 \vec{A}_\perp$:

$$\vec{v}_p = -\frac{m}{q B^4} \left( -B^2 \frac{d\vec{E}_\perp}{dt} \right) = \frac{m}{q B^2} \frac{d\vec{E}_\perp}{dt}$$

This is the polarization drift velocity. Notice the critical features:

• $\vec{v}_p$ is parallel to the changing electric field $\frac{d\vec{E}_\perp}{dt}$.
• $\vec{v}_p$ is proportional to particle mass $m$. Because $M_i \gg m_e$, polarization drift is overwhelmingly an ion phenomenon: $\vec{v}_{pi} \gg \vec{v}_{pe}$.
• $\vec{v}_p$ depends on charge $q$; ions and electrons drift in opposite directions, causing physical charge separation (polarization of the plasma).

1. The Polarization Current Density

Because positive ions and negative electrons drift in opposite directions along $\frac{d\vec{E}}{dt}$, the polarization drift creates a net macroscopic current density, termed the polarization current $\vec{J}_p$:

$$\vec{J}_p = n_0 e (\vec{v}_{pi} - \vec{v}_{pe}) = n_0 e \left( \frac{M_i}{e B^2}\frac{d\vec{E}}{dt} - \frac{m_e}{-e B^2}\frac{d\vec{E}}{dt} \right) = \frac{n_0(M_i + m_e)}{B^2}\frac{d\vec{E}}{dt}$$

Defining the total mass density $\rho_m \equiv n_0(M_i + m_e) \approx n_0 M_i$:

$$\vec{J}_p = \frac{\rho_m}{B^2} \frac{d\vec{E}}{dt}$$

2. Dielectric Permittivity & Plasma Capacitance

In Maxwell's Ampère-Maxwell law, the total transverse current is the sum of conduction current, polarization current, and vacuum displacement current:

$$\nabla \times \vec{B} = \mu_0 \left( \vec{J} + \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \right) = \mu_0 \left( \vec{J}_p + \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \right) = \mu_0 \left( \frac{\rho_m}{B^2} + \varepsilon_0 \right) \frac{\partial \vec{E}}{\partial t}$$

The combined term acts as an effective displacement current in a medium with effective permittivity $\varepsilon$:

$$\varepsilon \equiv \varepsilon_0 + \frac{\rho_m}{B^2} = \varepsilon_0 \left( 1 + \frac{\rho_m}{\varepsilon_0 B^2} \right)$$

Recalling the definition of the Alfvén speed $v_A \equiv \frac{B}{\sqrt{\mu_0 \rho_m}}$, and noting $c^2 = 1/(\mu_0 \varepsilon_0)$:

$$\frac{\rho_m}{\varepsilon_0 B^2} = \frac{\rho_m \mu_0 c^2}{B^2} = \frac{c^2}{v_A^2}$$

Thus, the low-frequency relative dielectric constant (dielectric permittivity) $\epsilon_r$ of a magnetized plasma is:

$$\epsilon_r = \frac{\varepsilon}{\varepsilon_0} = 1 + \frac{c^2}{v_A^2}$$

In typical laboratory fusion and space plasmas where $v_A \ll c$ (e.g., $v_A \sim 10^6\text{ m/s}$), the dielectric constant is enormous: $\epsilon_r \sim 10^4 \text{ to } 10^6$. The magnetized plasma acts as an immense energy storage capacitor.