5.1 Lorentz force

Definition of Lorentz force

The Lorentz force describes the total electromagnetic force on a charged particle due to both electric and magnetic fields. It governs how charges accelerate, curve, and drift in electromagnetic environments, making it central to everything from accelerator physics to plasma confinement.

The force is named after Hendrik Antoon Lorentz, who formulated the equation in the late 19th century. It unifies the electric and magnetic interactions into a single expression and serves as the starting point for analyzing charged particle dynamics in electrodynamics.

Force on a moving charge

The Lorentz force on a point charge is:

$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$

where $q$ is the particle's charge, $\vec{E}$ is the electric field, $\vec{v}$ is the particle's velocity, and $\vec{B}$ is the magnetic field. The equation is exact and relativistically correct when $\vec{E}$ and $\vec{B}$ are the fields in the frame where the particle has velocity $\vec{v}$.

Depending on the field configuration, this force can accelerate a particle along its direction of motion, deflect it transversely, or both.

Magnetic vs electric force

The two components of the Lorentz force behave quite differently:

• Electric force $q\vec{E}$: Acts on any charge, moving or stationary. It's parallel (or antiparallel) to $\vec{E}$ and can do work on the particle, changing its kinetic energy.
• Magnetic force $q\vec{v} \times \vec{B}$: Acts only on moving charges. It's always perpendicular to $\vec{v}$, so it changes the particle's direction but never its speed. The magnetic force does no work.

This distinction matters. In a purely magnetic field, a particle's kinetic energy is constant; only its trajectory changes. If you need to change a particle's speed, you need an electric field.

Mathematical formulation

The full vector equation $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ encodes both the magnitude and direction of the force. Working with it requires comfort with the cross product and careful attention to signs.

Vector cross product

The magnetic contribution involves the cross product $\vec{v} \times \vec{B}$:

• The result is a vector perpendicular to both $\vec{v}$ and $\vec{B}$, with direction given by the right-hand rule.
• Its magnitude is $|\vec{v} \times \vec{B}| = vB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.
• When $\vec{v} \parallel \vec{B}$, the cross product vanishes. When $\vec{v} \perp \vec{B}$, it's maximal.

In component form, if $\vec{v} = (v_x, v_y, v_z)$ and $\vec{B} = (B_x, B_y, B_z)$, then:

$\vec{v} \times \vec{B} = (v_yB_z - v_zB_y,\; v_zB_x - v_xB_z,\; v_xB_y - v_yB_x)$

Magnitude and direction

The total force magnitude is:

$|\vec{F}| = |q||\vec{E} + \vec{v} \times \vec{B}|$

You can't simply add $|q\vec{E}|$ and $|q\vec{v} \times \vec{B}|$ unless the two contributions happen to be parallel. In general, you need to compute the vector sum first, then take the magnitude.

For the direction: a positive charge is pushed along $\vec{E} + \vec{v} \times \vec{B}$. A negative charge is pushed opposite to that vector. This sign flip is the reason electrons and protons curve in opposite directions in a magnetic field.

Units of Lorentz force

The SI unit of the Lorentz force is the newton (N). A quick dimensional check confirms consistency:

$[q\vec{E}] = \text{C} \cdot \text{V/m} = \text{C} \cdot \frac{\text{J/C}}{\text{m}} = \frac{\text{J}}{\text{m}} = \text{N}$

$[q\vec{v}\times\vec{B}] = \text{C} \cdot \frac{\text{m}}{\text{s}} \cdot \text{T} = \text{C} \cdot \frac{\text{m}}{\text{s}} \cdot \frac{\text{kg}}{\text{A·s}^2} = \text{N}$

Both terms give newtons, as they must.

Magnetic field interactions

The magnetic force $q\vec{v} \times \vec{B}$ depends entirely on the relationship between the particle's velocity and the field direction. Decomposing $\vec{v}$ into components parallel and perpendicular to $\vec{B}$ is the standard approach.

Perpendicular velocity component

When $\vec{v} \perp \vec{B}$, the magnetic force reaches its maximum value:

$F = qvB$

This force is always perpendicular to $\vec{v}$, so it acts as a centripetal force, bending the particle into a circular orbit. Because the force never has a component along the velocity, it does no work and the particle's speed stays constant.

Parallel velocity component

When $\vec{v} \parallel \vec{B}$, the cross product $\vec{v} \times \vec{B} = 0$. There is no magnetic force at all, and the particle drifts freely along the field lines at constant velocity.

This is why you decompose: the parallel component is unaffected by $\vec{B}$, while the perpendicular component produces circular motion.

Stationary charges in magnetic fields

A charge at rest ($\vec{v} = 0$) experiences zero magnetic force. Only the electric force $q\vec{E}$ acts on it. This follows directly from the cross product vanishing when $\vec{v} = 0$, and it's consistent with the fact that the magnetic force can never do work.

Charged particle motion

The trajectory of a charged particle in given fields is found by solving Newton's second law with the Lorentz force: $m\dot{\vec{v}} = q(\vec{E} + \vec{v} \times \vec{B})$. The solutions depend on the field geometry and the particle's initial conditions.

Circular paths in uniform fields

For a particle moving perpendicular to a uniform $\vec{B}$ (with $\vec{E} = 0$), the magnetic force provides centripetal acceleration:

$qvB = \frac{mv^2}{r}$

Solving for the cyclotron radius (also called the Larmor radius):

$r = \frac{mv}{qB}$

The cyclotron frequency (angular) is:

$\omega_c = \frac{qB}{m}$

A key feature: $\omega_c$ is independent of the particle's speed. Faster particles orbit in larger circles but at the same frequency. This is what makes the cyclotron work.

Helical paths in uniform fields

When the velocity has both parallel and perpendicular components relative to $\vec{B}$, the motion is a helix. The perpendicular component $v_\perp$ produces circular motion with radius $r = mv_\perp / qB$, while the parallel component $v_\parallel$ carries the particle along the field at constant speed.

The pitch of the helix (distance advanced per revolution) is:

$p = v_\parallel T = v_\parallel \frac{2\pi m}{qB}$

Velocity selector applications

A velocity selector uses crossed $\vec{E}$ and $\vec{B}$ fields (perpendicular to each other and to the particle beam) to filter particles by speed. The condition for a particle to pass through undeflected is:

$qE = qvB \implies v = \frac{E}{B}$

Particles with this specific velocity experience zero net force transverse to their motion. Faster or slower particles are deflected and blocked. The selected velocity is independent of $q$ and $m$, which is why velocity selectors work for mixed beams.

Magnetic force on current-carrying wire

A current-carrying wire in a magnetic field feels a macroscopic force because the Lorentz force acts on each moving charge carrier inside the wire. Summing over all carriers in a length of wire gives a net force on the conductor.

Direction of force

The force on a straight current-carrying segment is:

$\vec{F} = I\vec{L} \times \vec{B}$

where $I$ is the current, $\vec{L}$ points along the wire in the direction of conventional current, and $\vec{B}$ is the external field.

To find the direction using the right-hand rule:

1. Point your fingers along $\vec{L}$ (direction of conventional current).
2. Curl them toward $\vec{B}$.
3. Your thumb points in the direction of $\vec{F}$.

Magnitude of force

The magnitude is:

$F = ILB\sin\theta$

where $\theta$ is the angle between the wire and the field.

• $\theta = 90°$: maximum force, $F = ILB$
• $\theta = 0°$ or $180°$: zero force (wire parallel to field)

For a curved wire, you integrate: $\vec{F} = I\int d\vec{l} \times \vec{B}$.

Magnetic torque on current loops

A planar current loop in a uniform magnetic field experiences no net force (the forces on opposite sides cancel), but it does experience a torque that tends to align the loop's magnetic dipole moment with the field.

The magnetic dipole moment of a loop is:

$\vec{\mu} = NIA\hat{n}$

where $N$ is the number of turns, $I$ is the current, $A$ is the loop area, and $\hat{n}$ is the unit normal to the loop (direction set by the right-hand rule with the current).

The torque is:

$\vec{\tau} = \vec{\mu} \times \vec{B}$

with magnitude $\tau = \mu B \sin\alpha$, where $\alpha$ is the angle between $\vec{\mu}$ and $\vec{B}$. The associated potential energy is $U = -\vec{\mu} \cdot \vec{B}$.

Hall effect

The Hall effect occurs when a current-carrying conductor is placed in a magnetic field perpendicular to the current. The Lorentz force deflects charge carriers to one side, building up a transverse voltage. This voltage, once it reaches steady state, directly reveals information about the carrier type, density, and the applied field.

Charge carrier drift

Here's the sequence of events:

1. Current flows along the conductor (say, in the $x$-direction), so carriers have a drift velocity $\vec{v}_d$.
2. An external magnetic field $\vec{B}$ is applied perpendicular to the current (say, in the $z$-direction).
3. The magnetic force $q\vec{v}_d \times \vec{B}$ pushes carriers sideways (in the $y$-direction).
4. Charge accumulates on one face of the conductor, creating a transverse electric field $\vec{E}_H$ (the Hall field).
5. Equilibrium is reached when the Hall electric force balances the magnetic force: $qE_H = qv_dB$.

The sign of the Hall voltage tells you the sign of the charge carriers. This is how you distinguish n-type from p-type semiconductors experimentally.

Hall voltage and Hall coefficient

At equilibrium, the Hall voltage across a conductor of thickness $t$ is:

$V_H = \frac{IB}{ntq}$

where $n$ is the carrier density, $t$ is the sample thickness (along $\vec{B}$), and $q$ is the carrier charge.

The Hall coefficient is defined as:

$R_H = \frac{E_y}{J_x B_z} = \frac{1}{nq}$

• For electrons ($q = -e$): $R_H < 0$
• For holes ($q = +e$): $R_H > 0$

Measuring $R_H$ gives you the carrier density directly, and combining it with conductivity measurements yields the carrier mobility $\mu = |R_H|\sigma$.

Hall effect applications

• Magnetic field sensors (Hall probes): Compact, solid-state devices used for position sensing, current measurement, and proximity detection. They output a voltage proportional to $B$.
• Semiconductor characterization: Hall measurements determine carrier type (n or p), carrier density $n$, and mobility $\mu$. This is a standard technique in materials science.
• Hall thrusters: Used in spacecraft propulsion. Electrons trapped in crossed $\vec{E}$ and $\vec{B}$ fields ionize propellant gas; the ions are then accelerated electrostatically to produce thrust.

Cyclotron motion

Cyclotron motion is the circular orbit of a charged particle in a uniform magnetic field. The cyclotron accelerator exploits the fact that the orbital frequency is independent of speed, allowing repeated acceleration with a fixed-frequency oscillating field.

Cyclotron frequency and radius

These are the same relations derived earlier, but they're worth restating in the accelerator context:

$\omega_c = \frac{qB}{m}, \qquad r = \frac{mv}{qB} = \frac{p}{qB}$

where $p = mv$ is the (non-relativistic) momentum. The frequency depends only on $q/m$ and $B$, not on the particle's energy. The radius grows linearly with speed.

Resonance condition

For a cyclotron to accelerate particles efficiently, the oscillating electric field across the gap between the dees must flip at exactly the cyclotron frequency:

$f_{\text{RF}} = f_c = \frac{qB}{2\pi m}$

When this resonance condition is satisfied, the particle arrives at the gap in phase with the accelerating field every half-turn, gaining energy $\Delta E = qV_0$ per crossing (where $V_0$ is the peak dee voltage).

If the particle becomes relativistic, its mass effectively increases ($m \to \gamma m$), the cyclotron frequency drops, and the fixed-frequency resonance breaks down. This is the fundamental energy limit of the classical cyclotron, and it's why synchrocyclotrons and synchrotrons were developed.

Principle of cyclotron accelerator

The operation of a cyclotron proceeds as follows:

1. A charged particle (e.g., a proton) is injected near the center of two hollow D-shaped electrodes ("dees") sitting in a uniform magnetic field $\vec{B}$ directed perpendicular to the dee plane.
2. The magnetic field bends the particle into a semicircular arc inside one dee.
3. When the particle reaches the gap between the dees, the oscillating voltage accelerates it across.
4. The particle enters the other dee with higher speed and therefore traces a larger semicircle.
5. This repeats every half-period. The particle spirals outward, gaining energy each time it crosses the gap.
6. At the outer edge, the particle is extracted by a deflector and directed to a target.

The maximum kinetic energy achievable is $K_{\max} = \frac{q^2B^2R^2}{2m}$, where $R$ is the maximum orbital radius (the dee radius).

Magnetic mirrors

Magnetic mirrors use spatially varying (inhomogeneous) magnetic fields to reflect charged particles, trapping them between regions of strong field. The underlying physics is the conservation of the magnetic moment (adiabatic invariant) of the particle's gyration.

Reflection of charged particles

As a particle spirals along a field line into a region of increasing $B$, the adiabatic invariant $\mu = \frac{mv_\perp^2}{2B}$ is conserved. Since $B$ increases, $v_\perp$ must increase. But total kinetic energy is conserved (the magnetic force does no work), so $v_\parallel$ must decrease.

If the field becomes strong enough, $v_\parallel \to 0$ and the particle reverses direction. This is the mirror reflection.

Whether a particle reflects or escapes depends on its pitch angle $\alpha$ at the midplane (where $B = B_0$). The pitch angle is defined by $\tan\alpha = v_\perp / v_\parallel$.

Loss cone angle

The critical pitch angle separating trapped from escaping particles is the loss cone angle $\theta_L$, given by:

$\sin^2\theta_L = \frac{B_0}{B_m}$

where $B_0$ is the field at the center and $B_m$ is the maximum field at the mirror points.

• Particles with pitch angle $\alpha > \theta_L$: reflected (trapped).
• Particles with pitch angle $\alpha < \theta_L$: escape through the mirror (lost).

The mirror ratio $R_m = B_m / B_0$ determines how tight the loss cone is. A large mirror ratio means a small loss cone and better confinement.

Magnetic mirror applications

• Plasma confinement in fusion devices: Early magnetic confinement schemes (e.g., magnetic mirror machines) used this principle. While simple mirrors suffer from end losses through the loss cone, tandem mirrors and other configurations partially address this.
• Van Allen radiation belts: Charged particles from the solar wind are trapped by Earth's dipole magnetic field, bouncing between mirror points near the poles. Particles in the loss cone precipitate into the atmosphere, producing aurorae.
• Particle traps: Penning traps combine a uniform magnetic field with an electrostatic quadrupole to confine single charged particles for high-precision measurements (e.g., electron g-factor, mass spectrometry).