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The Lorentz Force: How Fields Push Moving Charges

One rule governs every magnetic force on a charge — perpendicular, velocity-dependent, and, surprisingly, doing no work at all.

Only moving charges feel it

Here is the first surprise about the magnetic force. Park a charge at rest in a magnetic field and it feels nothing. Only when the charge moves does the field grab it. Move faster and the force grows in proportion; move straight along the field lines and the force vanishes entirely. The force depends on how fast the charge cuts across the field.

And the direction is downright strange. The magnetic force is perpendicular to both the velocity \vec{v} and the field \vec{B} at once — not along either of them. The mathematical object that produces a vector perpendicular to two others is the cross product, and the force law is written with it.

\vec{F} = q\,\vec{v}\times\vec{B}

The magnetic force on a charge q moving with velocity v in field B. The cross product makes F perpendicular to the plane of v and B.

The three directions of F = qv×B are mutually perpendicular. Point the fingers along v, curl toward B, and the thumb (for a positive charge) gives F.

Magnitude and the right-hand rule

F = qvB\sin\theta

The magnitude, where θ is the angle between v and B. Maximum force when the charge moves across the field (θ = 90°); zero when it moves along it (θ = 0).

To find the direction, use the right-hand rule for \vec{v}\times\vec{B}: point your fingers along \vec{v}, curl them toward \vec{B}, and your thumb points along \vec{v}\times\vec{B}. That is the force on a positive charge in the field. For a negative charge — an electron, say — the force points the opposite way, because q is negative.

A charge enters a uniform magnetic field and curves. Change its speed, the sign of its charge, and the field strength, and watch the force and path respond.

The force that does no work

Because the magnetic force is always perpendicular to the velocity, it never has a component along the motion — so it does zero work. By the work–energy theorem, a purely magnetic force can never change a particle's speed or kinetic energy. It only ever changes the direction of motion. A magnetic field is a perfect steering wheel, never an accelerator pedal.

Electric and magnetic, together

In the real world a charge often sits in both an electric field and a magnetic field. The total electromagnetic force adds the two contributions. This complete statement is the full Lorentz force — the single equation that, together with Newton's second law, dictates the motion of every charged particle in electromagnetism.

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

The full Lorentz force. The electric part qE can speed a charge up; the magnetic part qv×B only turns it.

Worked example: force on an electron

  1. Problem. An electron (q = -1.6\times10^{-19} C) moves at v = 2.0\times10^{6}\,\text{m/s} perpendicular to a uniform field B = 0.50 T. Find the magnitude and direction of the force.
  2. Magnitude. Since v ⟂ B, θ = 90° and sin θ = 1, so F = qvB = (1.6\times10^{-19})(2.0\times10^{6})(0.50).
  3. Compute. F = 1.6\times10^{-13}\,\text{N}. Small in everyday terms, but enormous for a particle whose weight is only about 10^{-29} N — the magnetic force utterly dominates gravity here.
  4. Direction. Apply the right-hand rule to v×B to get the direction for a positive charge, then reverse it because the electron's charge is negative. That reversed direction is where the electron is pushed, curving its path.