Moving Charges and Magnetism - Quick Revision
Magnetic force
- Lorentz force: F = q(E + v x B). The magnetic part F = q v B sin(theta) is perpendicular to both v and B.
- It is zero when v is parallel/antiparallel to B (theta = 0 or 180) and when v = 0; maximum when v is perpendicular to B.
- Magnetic force does no work (always perpendicular to v): speed and KE are unchanged, only direction changes.
- Unit of B is the tesla (T); 1 gauss = 10^-4 T.
Motion in a magnetic field
- For v perpendicular to B the path is a circle, radius r = mv/(qB).
- Cyclotron frequency omega = qB/m, f = qB/(2 pi m) - independent of speed and radius.
- If v has a component along B, the path is a helix; pitch p = v_parallel x T.
Force on a conductor
- F = I L x B, magnitude F = B I L sin(theta).
- Parallel currents attract, antiparallel currents repel; force per length f = mu0 I1 I2/(2 pi d) (defines the ampere).
Fields from currents
- Biot-Savart: dB = (mu0/4 pi) I dl sin(theta)/r^2; mu0/4 pi = 10^-7 T m/A.
- Long straight wire: B = mu0 I/(2 pi a).
- Centre of circular loop (N turns): B = mu0 N I/(2R).
- On the axis of a loop: B = mu0 I R^2 / [2(x^2 + R^2)^(3/2)].
- Inside a long solenoid: B = mu0 n I (n = turns per metre).
Torque and instruments
- Torque on a loop: tau = N I A B sin(theta) = m B sin(theta), with magnetic moment m = N I A.
- Stable equilibrium when m is parallel to B; no torque when m is parallel or antiparallel to B.
- A moving-coil galvanometer reads current via tau = N I A B; converted to an ammeter with a small parallel shunt and to a voltmeter with a large series resistance.
Common traps
- Magnetic force never changes speed/KE (does no work).
- r = mv/(qB) grows with momentum; frequency does NOT depend on speed.
- A current element produces zero field along its own direction (theta = 0).