Why Two Parallel Wires Carrying Current the Same Way Pull Together

Every current is surrounded by a magnetic field, and every magnetic field pushes on moving charges. Those two facts explain electromagnets, electric motors and the meters that measure current.

This lesson covers the Biot-Savart law for a loop, Ampere's law for wires and solenoids, magnetic forces on charges and conductors, and the torque that drives a galvanometer.

**The Biot-Savart law gives the field of a small current element as $dB=\frac{{\mu }_0}{4\pi }\frac{Idl\sin \theta }{r^2}$, which for a coil of N turns and radius R gives $B=\frac{{\mu }_{0}NI}{2R}$ at the centre and $B=\frac{{\mu }_{0}NI{R}^2}{2(R^2+x^2{)}^{3/2}}$ on the axis.

At the centre**, every element is at distance R and at $90^{\circ }$ to the line joining it to the centre, so the contributions simply add. The direction follows the right-hand thumb rule.

Worked example. A coil of 50 turns and radius 0.10 m carries 5.0 A:

$$B_{\text{centre}}=\frac{4\pi \times 10^{-7}\times 50\times 5.0}{2\times 0.10}=1.57\times 10^{-3}\text{ T}$$

On the axis at $x=0.10$ m, the factor $\frac{R^3}{(R^2+x^2{)}^{3/2}}=\frac{1}{2^{3/2}}=0.354$, so

$$B_{\text{axis}}=1.57\times 10^{-3}\times 0.354=5.55\times 10^{-4}\text{ T}$$

An everyday example. The coil inside an electric doorbell becomes a magnet whenever the button is pressed, pulling the striker towards the gong.

The substance. **Far along the axis, the loop's field falls as $1/x^3$**, exactly like a bar magnet's field — which is why a current loop behaves as a magnetic dipole.

**Ampere's circuital law states that $\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}$, which gives $B=\frac{{\mu }_{0}I}{2\pi r}$ around a long straight wire and $B={\mu }_{0}nI$ inside a long solenoid with n turns per metre.

Straight wire.** A circle of radius r centred on the wire has constant B along it, so $B\times 2\pi r={\mu }_{0}I$.

Solenoid. A rectangular loop with one side of length L inside the solenoid encloses $nLI$ of current. The field outside is nearly zero, so $BL={\mu }_{0}nLI$.

Worked example 1. At 0.050 m from a wire carrying 10 A:

$$B=\frac{4\pi \times 10^{-7}\times 10}{2\pi \times 0.050}=4.0\times 10^{-5}\text{ T}$$

Worked example 2. A solenoid with 1000 turns per metre carries 2.0 A:

$$B=4\pi \times 10^{-7}\times 1000\times 2.0=2.51\times 10^{-3}\text{ T}$$

An everyday example. An electric lock on a door uses a solenoid whose field pulls an iron bolt in or out.

The substance. Inside a long solenoid the field is uniform and does not depend on its radius — only on turns per metre and current.

**A charge moving in a magnetic field feels $F=qvB\sin \theta$, a conductor feels $F=BIl\sin \theta$, and two parallel wires a distance d apart feel $\frac{F}{l}=\frac{{\mu }_0{I}_1{I}_2}{2\pi d}$, attractive for currents in the same direction.

Moving charge.** The force is perpendicular to both v and B, so it does no work. A charge moving at right angles to B follows a circle of radius $r=\frac{mv}{qB}$.

Worked example 1. A proton at $2.0\times 10^6$ m s${}^{-1}$ enters a 0.50 T field at right angles:

$$F=1.6\times 10^{-19}\times 2.0\times 10^6\times 0.50=1.6\times 10^{-13}\text{ N}r=\frac{1.67\times 10^{-27}\times 2.0\times 10^6}{1.6\times 10^{-19}\times 0.50}=0.042\text{ m}$$

Worked example 2. A 0.20 m wire carrying 3.0 A across a 0.40 T field feels $F=0.40\times 3.0\times 0.20=0.24$ N.

Worked example 3 — parallel wires. Two wires 0.10 m apart each carry 5.0 A the same way:

$$\frac{F}{l}=\frac{4\pi \times 10^{-7}\times 5.0\times 5.0}{2\pi \times 0.10}=5.0\times 10^{-5}{\text{ N m}}^{-1}$$

An everyday example. A loudspeaker pushes its cone back and forth because the force on its current-carrying coil in a magnet's field reverses with the signal.

The substance. Each wire sits in the other's field — that is why like currents attract, the reverse of like charges.

**A coil of N turns and area A in a field B feels a torque $\tau =NIAB\sin \theta$; in a galvanometer this is balanced by a spring, giving a deflection proportional to current, and a small shunt converts it to an ammeter while a large series resistance converts it to a voltmeter.

Worked example 1 — torque.** A 50-turn coil of area 0.010 m${}^2$ carries 2.0 A in a 0.30 T field, with its plane along the field:

$$\tau =50\times 2.0\times 0.010\times 0.30=0.30\text{ N m}$$

Inside a galvanometer, a radial field keeps $\sin \theta =1$ at every position, so $NIAB=k\varphi$ and the scale is linear.

Worked example 2 — conversion. A galvanometer has $G=25\Omega$ and full-scale current $I_g=4.0$ mA.

- Ammeter for 0 to 2.0 A — a shunt in parallel: $S=\frac{I_{g}G}{I-I_g}=\frac{4.0\times 10^{-3}\times 25}{2.0-0.004}=0.050\Omega$
- Voltmeter for 0 to 10 V — a resistance in series: $R=\frac{V}{I_g}-G=\frac{10}{4.0\times 10^{-3}}-25=2475\Omega$

An everyday example. The needle of an analogue multimeter is a moving-coil galvanometer, switched between shunts and series resistors by its dial.

The substance. An ideal ammeter has near-zero resistance and an ideal voltmeter near-infinite resistance, so each disturbs the circuit as little as possible.