Magnetism, induction and Maxwell's equations

Physics · Heat, electricity and light. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A magnetic field acts on moving charges and on currents. The magnetic force is always at right angles to the velocity, so it bends the path but does no work and does not change the speed. Currents also produce fields: the Biot–Savart law gives the field of a small current element and Ampère's law gives the field of symmetric currents.

A changing magnetic flux through a circuit induces an e.m.f. (Faraday's law). The induced current always opposes the change that causes it (Lenz's law), which is conservation of energy.

Maxwell's equations collect these laws. Maxwell's own addition, the displacement current, shows that a changing electric field produces a magnetic field. Together they predict electromagnetic waves travelling at c.

Rules to remember

Common mistake

Thinking a magnetic field speeds up a charged particle. The force is perpendicular to the velocity, so the kinetic energy stays the same; only the direction changes.

Worked example

A proton (charge 1.6 × 10−19 C, mass 1.67 × 10−27 kg) moves at 2 × 106 m s−1 at right angles to a field of 0.5 T. Find the force on it and the radius of its path.

  1. F = qvB = 1.6 × 10−19 × 2 × 106 × 0.5 = 1.6 × 10−13 N.
  2. This force is the centripetal force: qvB = mv2/r, so r = mv/(qB).
  3. mv = 1.67 × 10−27 × 2 × 106 = 3.34 × 10−21 kg m s−1; qB = 1.6 × 10−19 × 0.5 = 8.0 × 10−20 C T.
  4. r = (3.34 × 10−21) ÷ (8.0 × 10−20) = 0.042 m.

Answer: F = 1.6 × 10−13 N; r ≈ 4.2 cm.

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