Electromagnetism · force and energy

Charged Particle

How can a magnetic field turn motion without changing speed? The magnetic force always points at right angles to the velocity and does no work. All it can do is turn the particle. An electric field is different: it can change the kinetic energy directly.

Send a particle through a cyclotron gap and tune the RF rhythm. Compare acceleration with phase slip, then switch to uniform fields to explore magnetic circles, drift and electric acceleration.

Speed $4\,\mathrm{m\,s^{-1}}$

Magnetic radius $2\,\mathrm{m}$

Kinetic energy $8\,\mathrm{J}$

Electric work $0\,\mathrm{J}$

A kick at the right moment

Keep the gap in step

Flight pathA faint full path accompanies the bright replay trail.Velocity and forceMint marks velocity; amber marks force between gap crossings.Field directionsBlue arrows show the normal field; gap arrows reverse with the RF voltage.Magnetic field

B_z

2\,\mathrm{T}

Injection speed

v_0

4\,\mathrm{m\,s^{-1}}

Gap voltage amplitude

V_0

16\,\mathrm{V}

RF frequency / cyclotron frequency

\omega_{\mathrm{RF}}/\omega_c

1

Injection phase

\phi_0

0

Horizontal electric field

E_x

0\,\mathrm{V\,m^{-1}}

Vertical electric field

E_y

0\,\mathrm{V\,m^{-1}}

Launch angle

\alpha

0^\circ

Charge

q

1\,\mathrm{C}

Mass

m

1\,\mathrm{kg}

Flight duration

T

12\,\mathrm{s}

Drag the mint launch handle horizontally to change injection speed. Sliders recalculate the flight immediately; replay and CSV use the same stored trajectory.

Model assumptions

The particle is nonrelativistic, and all quantities are in SI units. Uniform-field modes use fixed electric and magnetic fields. RF mode uses magnetic arcs and ideal gap kicks; collisions, radiation, space charge, relativity and back reaction are left out. The positive horizontal and vertical axes, together with the magnetic field normal to the plane, form a right-handed system. The view is scaled to fit the whole recorded flight.

Replay time

2\,\mathrm{s}

The latest gap crossings

Each energy step and each crossing marker come from the trajectory shown above.

Model, units and sources

The RF mode uses exact nonrelativistic magnetic arcs and a zero-width gap kick. The dees are shown with their roofs removed; the gap and particle are enlarged. Charge, mass, length, time and voltage use SI units at a teaching scale. The RF clock is fixed during each recorded flight. Electrode outlines resize to enclose the saved flight, so there is no fixed-aperture loss model. Fringing fields, finite gap transit time, collisions, radiation, space charge, relativistic phase slip and extraction are omitted. A decelerating barrier that would reflect the particle ends the trajectory at the gap.

\omega_c=\frac{|qB_z|}{m},\qquad \Delta K_n=qV_0\cos(\omega_{\mathrm{RF}}t_n+\phi_0)\,\mathrm{sgn}(v_{x,n})

The cyclotron angular frequency is set by charge, magnetic field and mass. At each arrival time, the signed gap voltage and crossing direction determine the work. In the uniform-field modes, electric work is calculated from displacement, and no RF gap acts on the particle.

Lawrence’s cyclotron lecture Lawrence and Livingston, 1932 Read the experimental history

Start with the essentials

Focus question: Why can a pure magnetic field bend a path into a circle without making the particle faster?
One-sentence intuition: Magnetic force stays perpendicular to velocity and does no work; the electric field directly changes kinetic energy.

Core mathematical model

Checking work against kinetic energy

K(t)-K(0)=q\,\mathbf E\cdot[\mathbf r(t)-\mathbf r(0)]

In a uniform field, the electric work depends only on the displacement. Compare it with the change in kinetic energy computed from the recorded velocity. The magnetic field does no work.

Arrive at the gap in phase

\omega_c=\frac{|qB_z|}{m},\qquad\Delta K=qV_0\cos(\omega_{\mathrm{RF}}t+\phi_0)\,\mathrm{sgn}(v_x)

A nonrelativistic particle takes the same time for each magnetic half-turn. Reversing the gap field at that rhythm adds energy at every crossing. Detuning eventually presents a decelerating phase. The gap supplies instantaneous work; transit time is omitted.

Lorentz force

\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)

The cross product sets both the size and the direction of the magnetic force. Flipping the sign of the charge reverses the total force.

Gyration radius and drift velocity

r_c=\frac{mv_\perp}{|q|B},\qquad\mathbf v_D=\frac{\mathbf E\times\mathbf B}{B^2}

A heavier particle or a faster perpendicular speed gives a larger orbit. The $\mathbf E\times\mathbf B$ drift does not depend on the mass or on the sign of the charge, as long as the charge and the magnetic field are nonzero.

Common difficulties

A magnetic field does not increase speed

Typical misconceptionA force that keeps acting must speed the particle up.

Better mental modelA force perpendicular to the velocity only turns it, so the power $q(\mathbf v\times\mathbf B)\cdot\mathbf v$ is always zero.

A negative charge reverses the right-hand rule

Typical misconception $\mathbf v\times\mathbf B$ is the force direction for every particle.

Better mental model $\mathbf v\times\mathbf B$ gives the force direction for a positive charge; when $q<0$ , the force points the other way.

Run the experiment

01
Compare the RF rhythm
Compare In step, Losing the rhythm and A quarter-cycle late with the recorded gap voltages.
What to observe: Resonance expands the orbit; detuning eventually removes energy; a quarter-cycle offset places the crossings at voltage zeros.
02
Start with pure circular motion
Choose Pure magnetic circle and drag through the recorded time.
What to observe: The speed stays the same, and the path bends into a circle.
03
Reverse the charge
Press Reverse charge, then compare the recorded paths.
What to observe: The particle circles the other way, with about the same radius.
04
Add a perpendicular electric field
Choose Crossed-field drift, move the electric-field slider.
What to observe: The center of the orbit starts to move steadily sideways.

Charged Particle

What happened at each gap crossing?

The latest gap crossings