The magnetic force on a moving charge
Oersted showed in 1820 that currents produce magnetic fields. A magnetic field B, in turn, exerts a force on a moving charge:
F = q (v × B), with magnitude F = qvB sin θ
The force is perpendicular to both v and B.
It is zero if the charge is at rest or moves parallel to B.
Because it is always perpendicular to the velocity, the magnetic force does no work and doesn't change the speed, only the direction.
With an electric field too, the total Lorentz force is F = q(E + v × B).
The SI unit of B is the tesla (T). The Earth's field is about 3.6 × 10⁻⁵ T.
Motion of a charge in a magnetic field
A charge moving perpendicular to a uniform field moves in a circle. The magnetic force provides the centripetal force:
qvB = mv²/r, so r = mv ⁄ qB
The angular frequency ω = qB/m and the frequency ν = qB/(2πm) do not depend on the speed or radius. Faster particles move in bigger circles but take the same time per revolution.
If the velocity has a component along the field as well, the path is a helix.
The cyclotron uses this fact to accelerate charged particles. Particles circle inside two hollow D-shaped electrodes ("dees") in a magnetic field. An alternating voltage at the cyclotron frequency qB/(2πm) accelerates them every time they cross the gap. Their radius grows with each turn while the timing stays in step. Cyclotrons are used to produce radioactive isotopes for medicine.
The Biot-Savart law
The magnetic field produced by a small element dl of a wire carrying current I, at a point at distance r:
dB = (μ₀ ⁄ 4π) I dl sin θ ⁄ r², directed along dl × r
μ₀ = 4π × 10⁻⁷ T m/A is the permeability of free space.
Field at the centre of a circular loop of radius R carrying current I:
B = μ₀I ⁄ 2R (for N turns, μ₀NI/2R)
On the axis of the loop at distance x from the centre: B = μ₀IR² ⁄ 2(x² + R²)^(3/2).
The right-hand thumb rule gives the direction: curl the fingers along the current, and the thumb points along the field through the loop.
Ampère's circuital law
The line integral of B around any closed loop equals μ₀ times the total current passing through the loop: ∮ B · dl = μ₀ I_enclosed
It is useful when the field has a high degree of symmetry.
Long straight wire: using a circular loop of radius r around the wire, B × 2πr = μ₀I, so
B = μ₀I ⁄ 2πr
The field lines are concentric circles around the wire.
Long solenoid with n turns per unit length:
B = μ₀nI inside, uniform and parallel to the axis; nearly zero outside.
Toroid (a solenoid bent into a ring) with N turns and radius r: B = μ₀NI/2πr inside its core, and zero in the open space inside and outside.
Force between two parallel currents
A wire carrying current I₁ produces a field at a parallel wire carrying I₂ at distance d. The force per unit length is
F/L = μ₀I₁I₂ ⁄ 2πd
Currents in the same direction attract.
Currents in opposite directions repel.
(This is the opposite of electric charges, where like charges repel.)
The ampere was historically defined from this force: 1 A is the current which, flowing in two parallel wires 1 m apart in vacuum, gives a force of 2 × 10⁻⁷ N per metre of length.
Torque on a current loop
A rectangular loop of area A carrying current I, with N turns, in a uniform field B experiences a torque:
τ = NIAB sin θ, or τ = m × B
where m = NIA is the loop's magnetic dipole moment, perpendicular to the plane of the loop. A current loop behaves like a bar magnet.
The moving coil galvanometer
A galvanometer detects and measures small currents. A coil suspended in a radial magnetic field (made by concave pole pieces and a soft iron core) experiences a torque NIAB, balanced by a spring's restoring torque kφ:
φ = (NAB ⁄ k) I
So the deflection φ is proportional to the current, giving a linear scale.
Current sensitivity = φ/I = NAB/k.
Voltage sensitivity = φ/V = NAB/(kR_G).
Conversions:
To make an ammeter, connect a small resistance (shunt) in parallel: S = I_g G ⁄ (I − I_g).
To make a voltmeter, connect a large resistance in series: R = V/I_g − G.
Summary
F = qvB sin θ; magnetic forces do no work; charges circle with r = mv/qB and ω = qB/m.
Biot-Savart: dB = (μ₀/4π) I dl sin θ/r²; loop centre B = μ₀I/2R.
Ampère: straight wire B = μ₀I/2πr; solenoid B = μ₀nI.
Parallel currents in the same direction attract: F/L = μ₀I₁I₂/2πd.
A galvanometer becomes an ammeter with a parallel shunt, or a voltmeter with a series resistance.