A circular loop of radius $r$ carries current $I_0$. The magnetic field at the centre of the loop due to a $60°$ arc is:

Biot-Savart law: $dB = \mu_0 I dl\sin\phi / 4\pi r^2$. Ampere\'s law: $\oint B\cdot dl = \mu_0 I_{enc}$. Direction: right-hand rule. Current-carrying conductor in field: $\vec{F} = I(\vec{L}\times\vec{B})$ (force on wire). Force between parallel wires: $F/L = \mu_0 I_1 I_2/2\pi d$ (attractive if same direction, repulsive if opposite). This defines the SI unit of Ampere.

Lorentz force: $\vec{F} = q\vec{v}\times\vec{B} = qvB\sin\theta$. Circular motion: $qvB = mv^2/r$ → $r = mv/qB$. Cyclotron frequency: $f = qB/2\pi m$ (independent of speed). Velocity selector (crossed E and B): $E = vB$ → $v = E/B$. Mass spectrometer: $r = mv/qB$ → different masses separate.

Magnetic dipole moment: $\vec{m} = NIA\hat{n}$ (for coil of N turns, area A). Torque: $\vec{\tau} = \vec{m}\times\vec{B} = mB\sin\theta$. Potential energy: $U = -\vec{m}\cdot\vec{B} = -mB\cos\theta$. Moving coil galvanometer: $\tau_{magnetic} = nBIA$, $\tau_{spring} = k\theta$ → $\theta = nBIA/k$ (deflection proportional to current).

Faraday\'s law: $\varepsilon = -d\Phi_B/dt$ where $\Phi_B = \int B\cdot dA$ (magnetic flux). Lenz\'s law: induced current opposes change in flux (ensures energy conservation). Motional EMF: $\varepsilon = Blv$ (conductor of length $l$ moving at $v$ perpendicular to $B$). Self-inductance: $\varepsilon = -L\, dI/dt$; $L = N\Phi/I$. Solenoid: $L = \mu_0 N^2 A/l = \mu_0 n^2 Al$. Energy stored: $U = \frac{1}{2}LI^2$.

A wire of length L carrying current I in a field B feels $F = BIL\sin\theta$, where $\theta$ is the angle between the wire and the field. The direction follows Fleming's left-hand rule. Between two parallel wires the force per unit length is $\frac{\mu_0 I_1 I_2}{2\pi d}$ — attractive when the currents flow the same way and repulsive when they oppose, which is the opposite of what students expect from charges, and is the definition on which the ampere itself was based.

A coil of N turns and area A carrying current I in a radial field experiences a torque $NBIA$, balanced by the restoring torque $k\phi$ of the suspension, giving $\phi = \frac{NBA}{k}I$ — a deflection directly proportional to current. Sensitivity rises with N, B and A, and falls with k. A galvanometer becomes an ammeter by adding a small shunt resistance in parallel, and a voltmeter by adding a large resistance in series.

Diamagnetic substances are weakly repelled and have no unpaired electrons, with relative permeability just below 1. Paramagnetic substances have unpaired electrons and are weakly attracted. Ferromagnetic substances contain domains that align strongly with the field, giving permeability in the thousands, and they retain magnetisation after the field is removed — the hysteresis that makes permanent magnets possible. Heating a ferromagnet past its Curie temperature destroys the domain alignment and it becomes paramagnetic.