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Derivation

Lorentz Force and Cyclotron Motion

Statement

For a point charge \( q \) of mass \( m \) moving with velocity \( \vec{v} \) through an electric field \( \vec{E} \) and a magnetic field \( \vec{B} \), the total electromagnetic force is \( \vec{F} = q\left(\vec{E} + \vec{v}\times\vec{B}\right) \). Taking this as the definition of the fields and inserting it into \( m\dot{\vec{v}}=\vec{F} \), we solve the equation of motion for uniform static fields and obtain a helical cyclotron trajectory: circular gyration at angular frequency \( \omega_c = qB/m \) and radius \( r_c = mv_\perp/(qB) \) in the plane perpendicular to \( \vec B \), superposed with uniform glide along \( \vec B \) and a charge-independent \( \vec{E}\times\vec{B}/B^2 \) drift.

Why it matters

The Lorentz force is the operational definition of the electromagnetic field: \( \vec E \) and \( \vec B \) are known only through the force \( q(\vec E+\vec v\times\vec B) \) they exert on a test charge. It is the one equation that feeds Maxwell's fields back into the dynamics of matter, closing classical electrodynamics.

Cyclotron motion is its cleanest consequence. It fixes the operating frequency of particle accelerators, sets the gyroradius that governs magnetic confinement of fusion plasmas, and — because \( \omega_c \) is independent of speed at low energy — is what makes the classical cyclotron work. The same \( \omega_c \) reappears quantum-mechanically as the spacing of Landau levels \( E_n=\hbar\omega_c(n+\tfrac12) \).

Assumptions
The fields \( \vec E,\vec B \) are externally prescribed, uniform in space and constant in time.If dropped, the equation of motion loses its constant coefficients; spatial gradients give grad-\( B \) and curvature drifts and a time-varying \( B \) induces \( \vec E \) by Faraday's law, so the clean helix survives only as a guiding-centre approximation valid when \( r_c\ll L \).
The charge is a structureless point: no spin, no finite size, no intrinsic moment.If dropped, a real current loop or spinning particle also feels a torque and a force \( -\nabla(\vec\mu\cdot\vec B) \); omitting this discards magnetic-moment dynamics and spin precession.
Motion is non-relativistic, \( v\ll c \), with constant mass \( m \).If dropped, momentum is \( \vec p=\gamma m\vec v \) and the gyrofrequency becomes \( \omega_c=qB/(\gamma m) \), falling with energy — precisely why the plain cyclotron fails and the synchrotron is needed.
Radiation reaction and collisions are neglected over the timescale of interest.If dropped, an accelerating charge radiates (Larmor / synchrotron), and the Abraham-Lorentz self-force makes the orbit slowly decay; collisions randomise the gyrophase, turning coherent gyration into cross-field diffusion.
Derivation
1
\[ \vec{F} = q\vec{E} + q\,\vec{v}\times\vec{B} \]
Superposition of the electrostatic force \( q\vec E \) (from the electric-field-from-Coulomb-law result) and the magnetic force \( q\vec v\times\vec B \) (from the Biot-Savart / Ampere-force result); Maxwell's equations are linear so the forces add. Taken as the definition of \( \vec E,\vec B \). A
2
\[ m\frac{d\vec{v}}{dt} = q\vec{E} + q\,\vec{v}\times\vec{B} \]
Newton's second law with constant mass, \( \vec F=m\,d\vec v/dt \), so \( \vec p=m\vec v \). This is the equation of motion to be integrated. A
3
\[ \frac{d}{dt}\!\left(\tfrac12 m v^2\right) = \vec v\cdot\vec F = q\,\vec v\cdot\vec E + q\,\vec v\cdot(\vec v\times\vec B) = q\,\vec v\cdot\vec E \]
Dot the equation of motion with \( \vec v \). The scalar triple product \( \vec v\cdot(\vec v\times\vec B)=0 \) because \( \vec v\times\vec B\perp\vec v \): the magnetic force does no work, and only \( \vec E \) changes the speed. B
4
\[ \vec B = B\,\hat{\mathbf z},\qquad \vec E = E_x\,\hat{\mathbf x} + E_z\,\hat{\mathbf z} \]
A uniform \( \vec B \) supplies a natural axis; align \( \hat{\mathbf z} \) with it, then rotate about \( \hat{\mathbf z} \) to remove any \( E_y \). No loss of generality. B
5
\[ \vec v\times\vec B = (v_y B)\,\hat{\mathbf x} - (v_x B)\,\hat{\mathbf y} \]
Cross product of \( \vec v=(v_x,v_y,v_z) \) with \( B\hat{\mathbf z} \); the \( \hat{\mathbf z} \) component vanishes since \( \hat{\mathbf z}\times\hat{\mathbf z}=\vec 0 \). A
6
\[ \dot v_x = \frac{q}{m}E_x + \frac{qB}{m}v_y,\qquad \dot v_y = -\frac{qB}{m}v_x,\qquad \dot v_z = \frac{q}{m}E_z \]
Take components of Step 2 with Step 5 substituted. The axes decouple into a \( 2\times2 \) rotational block \( (x,y) \) and a free \( 1 \)-D block \( z \). B
7
\[ \omega_c \equiv \frac{qB}{m} \]
The combination \( qB/m \) has units of inverse time and is the only frequency in the problem; naming it is a definition. Signed here — the observed gyration rate is \( |\omega_c| \) and its sign fixes the handedness. A
8
\[ \dot v_z = \frac{q}{m}E_z \;\Rightarrow\; v_z(t) = v_{z0} + \frac{qE_z}{m}\,t \]
The parallel equation is constant acceleration; integrate once. Motion along \( \vec B \) is untouched by the magnetic force. A
9
\[ u_x \equiv v_x - v_d,\qquad v_d \equiv -\frac{E_x}{B} \]
Shift to the frame drifting at \( v_d \) to absorb the constant \( E_x \) term: since \( \omega_c v_d = -qE_x/(mB)\cdot B/1 = -(q/m)E_x \) cancels the drive, the perpendicular block becomes homogeneous. C
10
\[ \dot u_x = \omega_c v_y,\qquad \dot v_y = -\omega_c u_x \]
Substitute Step 9 into Step 6. Check: \( -\omega_c u_x = -\omega_c(v_x + E_x/B) = -\omega_c v_x \) because \( \omega_c E_x/B = qE_x/m \) exactly cancels the electric term of \( \dot v_y \)'s partner. The block is now pure rotation. C
11
\[ w \equiv u_x + i\,v_y \;\Rightarrow\; \dot w = \omega_c v_y - i\omega_c u_x = -i\omega_c\,w \]
Form the complex combination \( w \): adding the first equation of Step 10 to \( i \) times the second diagonalises the \( 2\times2 \) rotation into one linear first-order complex ODE. C
12
\[ w(t) = w_0\,e^{-i\omega_c t},\qquad |w| = v_\perp = \text{const} \]
Integrate the constant-coefficient linear ODE. The modulus is fixed, so the perpendicular speed \( v_\perp=|w_0| \) is conserved — consistent with Step 3. C
13
\[ u_x = v_\perp\cos(\omega_c t-\phi),\qquad v_y = -v_\perp\sin(\omega_c t-\phi) \]
Take real and imaginary parts of Step 12 with \( w_0=v_\perp e^{i\phi} \). The velocity vector rotates uniformly at rate \( \omega_c \); the sign of \( q \) sets the sense. B
14
\[ x(t) = v_d t + \frac{v_\perp}{\omega_c}\sin(\omega_c t-\phi) + x_g,\qquad y(t) = \frac{v_\perp}{\omega_c}\cos(\omega_c t-\phi) + y_g \]
Integrate \( v_x=u_x+v_d \) and \( v_y \) once. The transverse coordinates trace a circle of radius \( r_c\equiv v_\perp/\omega_c = mv_\perp/(qB) \) about the guiding centre \( (x_g,y_g) \), which itself glides at \( v_d \). B
15
\[ v_d = -\frac{E_x}{B} \;\Rightarrow\; \vec v_d = \frac{\vec E\times\vec B}{B^2} \]
Recast the scalar drift in vector form: with \( \vec E=E_x\hat{\mathbf x} \), \( \vec B=B\hat{\mathbf z} \), one has \( (\vec E\times\vec B)/B^2 = E_x(\hat{\mathbf x}\times\hat{\mathbf z})/B = -(E_x/B)\hat{\mathbf y} \). The drift is independent of \( q \), \( m \) and sign. C
Result
\[ \vec{F} = q\left(\vec{E}+\vec{v}\times\vec{B}\right),\qquad \omega_c=\frac{qB}{m},\qquad r_c=\frac{mv_\perp}{qB},\qquad \vec v_{E\times B}=\frac{\vec E\times\vec B}{B^2} \]

Reading. A charge feels an electric force along \( \vec E \) that does work, plus a magnetic force perpendicular to both \( \vec v \) and \( \vec B \) that does none. In a uniform \( \vec B \) the path is a helix: uniform glide along the field at \( v_\parallel \), circular gyration across it at frequency \( \omega_c \) (independent of speed) and radius \( r_c \) (set by the transverse momentum \( mv_\perp \)). A perpendicular \( \vec E \) adds a uniform \( \vec E\times\vec B \) drift, identical for every charge regardless of sign or mass, so it drives no net current.

Units check. \( [q\,\vec v\times\vec B] = \mathrm{C}\cdot(\mathrm{m\,s^{-1}})\cdot\mathrm{T} \); using \( \mathrm{T}=\mathrm{kg\,s^{-1}\,C^{-1}} \) gives \( \mathrm{C\,m\,s^{-1}}\cdot\mathrm{kg\,s^{-1}C^{-1}}=\mathrm{kg\,m\,s^{-2}}=\mathrm N \). For \( \omega_c \): \( \mathrm{C\cdot T/kg}=\mathrm{C\cdot kg\,s^{-1}C^{-1}/kg}=\mathrm{s^{-1}} \), an angular frequency. For \( r_c \): \( \mathrm{kg\,(m\,s^{-1})/(C\,T)}=\mathrm{kg\,m\,s^{-1}}/(\mathrm{kg\,s^{-1}})=\mathrm m \). Drift: \( [\vec E\times\vec B/B^2]=(\mathrm{V\,m^{-1}}\cdot\mathrm T)/\mathrm{T^2}=\mathrm{V\,m^{-1}\,T^{-1}}=\mathrm{m\,s^{-1}} \). All consistent.

Limiting cases
  • \( \vec B=\vec 0 \): motion reduces to uniform acceleration \( \vec a=q\vec E/m \) along \( \vec E \) — the electrostatic parabola.
  • \( \vec E=\vec 0,\ v_\parallel=0 \): a pure circle at \( \omega_c \), radius \( r_c=mv_\perp/(qB) \); the helix degenerates.
  • \( v_\perp=0 \) (\( \vec v\parallel\vec B \)): \( \vec v\times\vec B=\vec 0 \), the charge streams straight along \( \vec B \) undeflected.
  • Strong field / large \( \omega_c \): \( r_c\to 0 \), the particle is tied tightly to a field line — the guiding-centre limit of a magnetised plasma.
  • \( E_x=vB \) in crossed fields: the electric and magnetic transverse forces cancel and the charge glides straight — the velocity-selector condition.
Breaks when
  • Relativistic speeds. When \( v\sim c \), momentum is \( \gamma m\vec v \) not \( m\vec v \); the cyclotron frequency becomes \( \omega_c=qB/(\gamma m) \) and depends on energy, forcing isochronous cyclotrons to ramp the RF or shape \( B(r) \). Synchrotron radiation also removes energy each turn.
  • Non-uniform or time-varying fields. Gradients of \( B \) add grad-\( B \) and curvature drifts \( \sim \tfrac12 v_\perp r_c\,(\vec B\times\nabla B)/B^2 \); a changing \( B \) induces \( \vec E \) via Faraday's law. The closed helix is then only leading order, valid when \( r_c\ll L \) (field scale length).
  • Radiation reaction and collisions. The Larmor / Abraham-Lorentz self-force makes the orbit decay; collisions randomise the gyrophase, converting coherent gyration into cross-field diffusion. Both are neglected in the ideal single-particle result.
  • Quantum regime. When \( \hbar\omega_c \) is comparable to the transverse energy, orbits quantise into Landau levels \( E_n=\hbar\omega_c(n+\tfrac12) \); the classical continuum of radii no longer applies.
Failure modes
  • "The magnetic force does work." Claiming \( \vec F\cdot\vec v\neq0 \) for the magnetic term. It is identically zero; only \( \vec E \) changes speed. Energy given to a particle in a cyclotron comes from the accelerating \( \vec E \) in the gap, not from \( \vec B \).
  • Frequency depends on speed. Asserting faster particles gyrate faster. Non-relativistically \( \omega_c=qB/m \) is independent of \( v \); only the radius \( r_c\propto v_\perp \) grows. This is exactly what makes the classical cyclotron work.
  • Sign / handedness errors. Dropping the sign of \( q \) or mis-evaluating \( \vec v\times\vec B \) reverses the sense of gyration; electrons and ions circulate oppositely about the same \( \vec B \).
  • Charge-dependent \( \vec E\times\vec B \) drift. Writing the drift with a factor of \( q \). It is \( \vec E\times\vec B/B^2 \), independent of \( q \) and \( m \), so it separates no charge and drives no current — unlike grad-\( B \) drift.
  • Using \( |\vec v| \) instead of \( v_\perp \) in \( r_c \). Only the component perpendicular to \( \vec B \) sets the gyroradius; the parallel component fixes the helix pitch, not the circle.
  • Confusing \( f_c \) and \( \omega_c \). \( \omega_c=qB/m \) is angular; the ordinary frequency is \( f_c=\omega_c/(2\pi) \). Dropping the \( 2\pi \) is a factor-6 error.
Discussion

The Lorentz force welds the two halves of electromagnetism into one statement about motion: \( \vec E \) and \( \vec B \) are defined by the force \( q(\vec E+\vec v\times\vec B) \) they impose on a test charge. This is a companion postulate to Maxwell's equations, not a consequence of them — Maxwell's equations tell the fields how to evolve given charges and currents, while the Lorentz force tells charges how to move given the fields. Together they close the dynamics of classical electrodynamics.

The velocity-dependence of the magnetic term is what makes it special. Because \( \vec F\perp\vec v \), it is a pure curvature force: it bends the path without changing the speed, exactly as string tension curves a whirling stone without spinning it faster. The gyrofrequency \( \omega_c=qB/m \) being independent of speed is a small miracle of the linear structure — it is why Lawrence's cyclotron could use a fixed-frequency oscillator, and why it fails once relativity sends \( m\to\gamma m \).

The decomposition into guiding-centre drift plus gyration is the gateway to plasma physics. In the drift frame the transverse electric force is transformed away, revealing that uniform crossed \( \vec E,\vec B \) simply carry every particle sideways at \( \vec E\times\vec B/B^2 \), independent of charge and mass. Being charge-independent, this drift separates no charge and drives no current — central to magnetic confinement, where one must instead reckon with the charge-dependent grad-\( B \) and curvature drifts that do separate charge and set up polarisation fields.

Geometrically the whole result is one Lorentz-covariant statement: the four-force is \( \frac{dp^\mu}{d\tau}=qF^{\mu\nu}u_\nu \) with the antisymmetric field tensor \( F^{\mu\nu} \) housing \( \vec E \) and \( \vec B \). Antisymmetry of \( F^{\mu\nu} \) is precisely why \( u_\mu\,dp^\mu/d\tau = qF^{\mu\nu}u_\nu u_\mu = 0 \), the covariant restatement of "the magnetic force does no work" and of conservation of rest mass. The non-relativistic helix is the low-velocity shadow of this geometry; the constancy of \( \omega_c/\gamma \) and the onset of synchrotron radiation are its relativistic corrections.

Common misconceptions. That \( \vec B \) accelerates particles (it never does work), that the cyclotron radius and frequency scale the same way with speed (only the radius does), that the \( \vec E\times\vec B \) drift depends on the sign of the charge (it does not), and that the Lorentz force is a theorem of Maxwell's equations rather than an independent law defining the fields.

Worked examples
1
Proton in a fusion-relevant field: \( B=2.0\ \mathrm T \), \( \vec E=\vec 0 \), perpendicular speed \( v_\perp=1.0\times10^{6}\ \mathrm{m\,s^{-1}} \). Find \( \omega_c \), \( f_c \) and \( r_c \). Data: \( q=1.60\times10^{-19}\ \mathrm C \), \( m=1.67\times10^{-27}\ \mathrm{kg} \).
\[ \omega_c=\frac{qB}{m}=\frac{(1.60\times10^{-19})(2.0)}{1.67\times10^{-27}} \]
Insert numbers into \( \omega_c=qB/m \).
\[ \omega_c = 1.92\times10^{8}\ \mathrm{rad\,s^{-1}},\qquad f_c=\frac{\omega_c}{2\pi}=3.05\times10^{7}\ \mathrm{Hz}\approx 30.5\ \mathrm{MHz} \]
Divide by \( 2\pi \) for the ordinary frequency.
\[ r_c=\frac{v_\perp}{\omega_c}=\frac{1.0\times10^{6}}{1.92\times10^{8}}=5.2\times10^{-3}\ \mathrm m \]
Gyroradius from \( r_c=v_\perp/\omega_c \).
\[ \omega_c\approx1.9\times10^{8}\ \mathrm{rad\,s^{-1}},\quad f_c\approx31\ \mathrm{MHz},\quad r_c\approx5.2\ \mathrm{mm} \]

Reading. The proton gyrates about 30 million times a second on a 5 mm circle — far smaller than a tokamak, so the particle is well magnetised. The ion-cyclotron frequency lands in the RF band exploited for plasma heating.

2
Crossed-field velocity selector. A beam of ions crosses perpendicular \( \vec E \) and \( \vec B \) with \( E=3.0\times10^{4}\ \mathrm{V\,m^{-1}} \) and \( B=0.15\ \mathrm T \). Find the selected (undeflected) speed and the \( \vec E\times\vec B \) drift speed; comment.
\[ q\vec E + q\,\vec v\times\vec B = \vec 0 \;\Rightarrow\; E = vB \;\Rightarrow\; v=\frac{E}{B} \]
Zero net transverse force requires the electric and magnetic forces to cancel; magnitudes give \( E=vB \).
\[ v=\frac{3.0\times10^{4}}{0.15}=2.0\times10^{5}\ \mathrm{m\,s^{-1}} \]
Insert numbers.
\[ v_{E\times B}=\frac{E}{B}=2.0\times10^{5}\ \mathrm{m\,s^{-1}} \]
The drift magnitude is also \( E/B \); a particle at the drift speed sits in the zero-force frame.
\[ v_{\text{selected}}=v_{E\times B}=\frac{E}{B}=2.0\times10^{5}\ \mathrm{m\,s^{-1}} \]

Reading. Only ions with \( v=E/B \) pass straight through, independent of charge and mass — exactly the charge-independence of the \( \vec E\times\vec B \) drift. Slower ions over-deflect toward \( \vec E \), faster ions toward the magnetic force; this Wien filter feeds a mass spectrometer.

Problems
  1. An electron (\( m=9.11\times10^{-31}\ \mathrm{kg} \), \( |q|=1.60\times10^{-19}\ \mathrm C \)) moves in \( B=0.50\ \mathrm T \). Compute its cyclotron frequency \( f_c \).
    Solution \( \omega_c=|q|B/m=(1.60\times10^{-19})(0.50)/(9.11\times10^{-31})=8.78\times10^{10}\ \mathrm{rad\,s^{-1}} \). Then \( f_c=\omega_c/2\pi=1.40\times10^{10}\ \mathrm{Hz}=14.0\ \mathrm{GHz} \) (X/Ku-band — the basis of electron-cyclotron-resonance heating).
  2. A proton enters \( B=1.2\ \mathrm T \) at speed \( 3.0\times10^{5}\ \mathrm{m\,s^{-1}} \), pitched \( 30^\circ \) to \( \vec B \). Find the gyroradius and the helix pitch (advance per turn).
    Solution \( v_\perp=v\sin30^\circ=1.5\times10^{5} \), \( v_\parallel=v\cos30^\circ=2.60\times10^{5}\ \mathrm{m\,s^{-1}} \). \( \omega_c=qB/m=(1.60\times10^{-19})(1.2)/(1.67\times10^{-27})=1.15\times10^{8}\ \mathrm{rad\,s^{-1}} \). \( r_c=v_\perp/\omega_c=1.5\times10^{5}/1.15\times10^{8}=1.30\times10^{-3}\ \mathrm m=1.3\ \mathrm{mm} \). Period \( T=2\pi/\omega_c=5.47\times10^{-8}\ \mathrm s \); pitch \( p=v_\parallel T=2.60\times10^{5}\times5.47\times10^{-8}=1.42\times10^{-2}\ \mathrm m\approx14\ \mathrm{mm} \).
  3. Show that in a pure magnetic field the kinetic energy is exactly conserved, and state what physically supplies energy in a cyclotron accelerator.
    Solution From \( m\,d\vec v/dt=q\,\vec v\times\vec B \), dot with \( \vec v \): \( \tfrac{d}{dt}(\tfrac12 mv^2)=q\,\vec v\cdot(\vec v\times\vec B)=0 \) since \( \vec v\times\vec B\perp\vec v \). Hence \( |\vec v| \) is constant and KE is conserved. In a cyclotron the energy comes from the oscillating electric field in the accelerating gap between the dees, applied in phase with the gyration at \( f_c \); \( \vec B \) merely curves the path back for the next kick.
  4. In crossed fields \( \vec E=E\hat{\mathbf y} \), \( \vec B=B\hat{\mathbf z} \), a charge starts from rest at the origin. Using \( \vec v=\vec v_{E\times B}+\vec v_{\text{gyro}} \), describe the trajectory and give the maximum \( y \)-excursion.
    Solution Drift \( \vec v_{E\times B}=\vec E\times\vec B/B^2=(E\hat{\mathbf y}\times B\hat{\mathbf z})/B^2=(E/B)\hat{\mathbf x} \). In the drift frame the charge starts with velocity \( -(E/B)\hat{\mathbf x} \) and simply gyrates at \( \omega_c \) with speed \( E/B \), radius \( r=(E/B)/\omega_c=mE/(qB^2) \). In the lab frame this is a cycloid: net drift along \( +\hat{\mathbf x} \) with looping motion. The maximum \( y \)-excursion is the cycloid height \( 2r=2mE/(qB^2) \), reached each period; the charge returns to \( y=0 \) with zero \( y \)-velocity, confirming \( \vec B \) does no net work.
  5. A singly-charged ion of unknown mass is accelerated through \( V=2.0\ \mathrm{kV} \), then enters \( B=0.40\ \mathrm T \) perpendicular to its velocity and follows a circle of radius \( r=8.0\ \mathrm{cm} \). Find the ion mass and identify it. (\( q=1.60\times10^{-19}\ \mathrm C \).)
    Solution Energy: \( qV=\tfrac12 mv^2\Rightarrow v=\sqrt{2qV/m} \). Radius: \( r=mv/(qB)\Rightarrow v=qBr/m \). Eliminate \( v \): \( qBr/m=\sqrt{2qV/m}\Rightarrow q^2B^2r^2/m^2=2qV/m\Rightarrow m=\dfrac{qB^2r^2}{2V} \). Numerically \( m=(1.60\times10^{-19})(0.40)^2(0.080)^2/(2\times2000)=(1.60\times10^{-19})(0.16)(6.4\times10^{-3})/4000=4.1\times10^{-26}\ \mathrm{kg} \). In atomic mass units \( m=4.1\times10^{-26}/1.66\times10^{-27}\approx25\ \mathrm u \) — consistent with a light ion such as \( \mathrm{^{24}Mg^+} \), illustrating the mass-spectrometer principle.