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Magnetic Force

Source lecture(s): SC134 Lec9

Intuition

Only moving charges feel magnetic forces, and the force is perpendicular to both velocity and field.

Formal Definition

Lorentz force: \(\vec{F} = q\vec{v}\times\vec{B}\)

Mathematical Formulation

Magnitude: \(F = qvB\sin\theta\) Direction: right-hand rule. Cyclotron radius: \(r = mv/(|q|B)\).

Derivation

Cross product follows from relativistic transformation of electric fields; for current elements integrate \(dq\vec{v}\).

Worked Example

Electron (\(v=10^6\,\text{m/s}\), \(B=0.1\,\text{T}\)) circles with \(r\approx 5.7\times10^{-5}\,\text{m}\).

Common Mistakes

  • Using electric force formula for magnetic fields.
  • Getting sign wrong for electrons.

Quiz

Q1. Can a magnetic field change a particle's speed?

Answer

No—force is perpendicular to velocity.