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Electromagnetism · Lorentz force

Charged Particle

A particle moves under the Lorentz force F=q(E+v×B)\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B). The teal arrow is velocity and the amber arrow is force. A pure magnetic field curves the path into a circle; add an electric field and the circle drifts sideways.

Interactive modelCharged Particle
Speed0.00 m/s
Cyclotron radius∞
Cyclotron frequency0.00 rad/s
Kinetic energy0.00 J

Physics tutorial

Charged particles: electric fields change energy, magnetic fields turn motion

BackgroundA charged particle responds to electric and magnetic fields. Electric force follows E, while magnetic force comes from the cross product of velocity and B.

Why it mattersMass spectrometers, cyclotrons, plasma confinement, and the Hall effect all use the same Lorentz force.

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

Lorentz force

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

The cross product sets magnetic magnitude and direction; charge sign reverses the total force.

Gyration and drift scales

rc=mv⊥∣q∣B,vD=E×BB2r_c=\frac{mv_\perp}{|q|B},\qquad\mathbf v_D=\frac{\mathbf E\times\mathbf B}{B^2}

Mass and perpendicular speed enlarge the orbit; E×B\mathbf E\times\mathbf B drift is independent of mass and charge sign.

Common difficulties

A magnetic field does not increase speed

Typical misconceptionAny sustained force must make the particle faster.

Better mental modelA perpendicular force rotates velocity; power q(v×B)⋅vq(\mathbf v\times\mathbf B)\cdot\mathbf v is always zero.

Negative charge reverses the right-hand result

Typical misconceptionv×B\mathbf v\times\mathbf B is the force direction for every particle.

Better mental modelv×B\mathbf v\times\mathbf B gives the direction for positive charge; q<0q<0 reverses it.

Run the experiment

  1. 01

    Establish pure gyration

    Choose Cyclotron and set E to zero.

    What to observe: Speed stays fixed while the path becomes circular.
  2. 02

    Reverse one sign

    Reverse only q or Bz.

    What to observe: Rotation reverses while radius stays similar.
  3. 03

    Add a perpendicular electric field

    Choose E x B drift and vary E/B.

    What to observe: The orbit center develops steady sideways motion.