Sources of the magnetic field: Biot-Savart, Ampère's law and the solenoid

In the previous lesson the magnetic field was given. Here we ask where it comes from: the sources of the magnetic field are currents. We cover the Biot-Savart law, the field of a wire and of a loop, the force between two wires, Ampère’s law and the solenoid, up to the inductance we will use in the lesson on induction.

The Biot-Savart law

A wire element carrying current produces, at a point a distance away, the field

where T·m/A is the permeability of free space (since 2019 its value is measured rather than fixed, but it is the same to within one part in a billion). It is the counterpart of Coulomb’s law, with two differences: the source is a current element, not a charge, and the field is perpendicular to both and .

The field of a straight wire

Integrating Biot-Savart over an infinite wire (or, faster, with Ampère’s law below):

At 1 cm from a wire carrying 10 A the field is mT, four times the Earth’s field: enough to deflect a compass, as Ørsted saw.

At the centre of a circular loop of radius : each element is perpendicular to the radius and contributes along the axis, so .

The force between two wires and the definition of the ampere

Two parallel wires a distance apart, carrying and : wire 1 creates at wire 2 the field , and wire 2 feels the force . The force per unit length is

attractive if the currents flow the same way, repulsive if opposite. With 1 A in each wire 1 m apart the force is N/m: the old definition of the ampere. In short circuits these forces become huge: with tens of thousands of amperes, the busbars in switchboards must be braced to withstand forces of tonnes.

Ampère’s law

The circulation of the magnetic field around a closed line equals times the enclosed current:

It checks out at once on the wire: around a circle of radius , is constant and tangent, so . It plays the role Gauss’s law plays for the electric field: it always holds (for steady currents), but it helps compute the field only when there is symmetry. It is also the counterexample of Analysis 2: the wire’s field is irrotational outside the wire, but not conservative, because there is a “hole” where the current flows. In local form: .

The solenoid

A solenoid is a wire wound as a helix with turns per unit length. Inside, the fields of the turns add up and the field is nearly uniform and parallel to the axis; outside it almost vanishes.

Proof

Apply Ampère’s law to the rectangle in the figure, with one side of length inside the solenoid and one outside. Outside ; on the sides perpendicular to the axis : only the inside side remains, . The enclosed current is that of the turns crossed, . So

independent of position inside the solenoid. With 1000 turns per metre and 1 A, mT. With an iron core, whose relative permeability is in the thousands, the field grows by the same factor: the electromagnet of relays, solenoid valves and electric locks.

Inductance

The magnetic flux linked with a circuit is proportional to its current. The constant of proportionality is the inductance:

For a solenoid of turns, length and cross-section : the flux through one turn is , the flux linked with all turns is , so

Inductance grows with the square of the number of turns. Just as a capacitor stores energy in the electric field, an inductor stores it in the magnetic field, with energy and density . In the next lesson we will see what happens when the current changes: that is where inductance takes centre stage.