Why a Compass Needle Swings Near a Current-Carrying Wire

A compass placed near a wire carrying current swings away from north. That simple observation shows that moving charges produce magnetic fields and feel magnetic forces — the basis of electric motors, loudspeakers and particle accelerators.

This part covers Oersted's experiment and the force on moving charges, circular motion in a magnetic field, the force on a current-carrying conductor, and the Biot-Savart law.

**Oersted's experiment shows that a current-carrying wire deflects a nearby compass needle, so electric currents produce magnetic fields; a charge $q$ moving with velocity $\vec{v}$ in fields $\vec{E}$ and $\vec{B}$ feels the Lorentz force $\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$.

The experiment. A compass needle below a straight wire points north-south. When current flows, the needle turns; reversing the current turns it the other way; moving the needle above the wire also reverses the deflection. The magnetic field forms circles around the wire.

Magnetic force:**

$${\vec{F}}_B=q(\vec{v}\times \vec{B}),F_B=qvB\sin \theta$$

- Zero when the charge is at rest or moves parallel to $\vec{B}$
- Maximum, $qvB$, when $\vec{v}$ is perpendicular to $\vec{B}$
- Always perpendicular to the velocity, so it does no work

Worked example. A proton moves at $2\times 10^6$ m s${}^{-1}$ perpendicular to a $0.5$ T field:

$$F=qvB=1.6\times 10^{-19}\times 2\times 10^6\times 0.5=1.6\times 10^{-13}\text{ N}$$

An everyday example. A compass held close to thick household wiring carrying current can point the wrong way, showing the magnetic field of the current.

The substance. A magnetic force can change a charge's direction but never its speed, because it does no work.

**A charge moving perpendicular to a uniform magnetic field follows a circle, because the magnetic force $qvB$ provides the centripetal force; its radius is $r=\frac{mv}{qB}$ and its period is $T=\frac{2\pi m}{qB}$, which does not depend on speed.

Derivation:**

$$qvB=\frac{m{v}^2}{r}\Rightarrow r=\frac{mv}{qB},T=\frac{2\pi r}{v}=\frac{2\pi m}{qB}$$

Worked example. An electron of mass $9.1\times 10^{-31}$ kg moves at $3\times 10^7$ m s${}^{-1}$ perpendicular to a field of $10^{-3}$ T:

$$r=\frac{9.1\times 10^{-31}\times 3\times 10^7}{1.6\times 10^{-19}\times 10^{-3}}=\frac{2.73\times 10^{-23}}{1.6\times 10^{-22}}\approx 0.17\text{ m}$$

At an angle to the field. The velocity component along $\vec{B}$ is unchanged, while the perpendicular component produces circular motion, so the path is a helix with pitch $v_{\parallel }T$.

An everyday example. The glowing lights seen in polar skies come from charged particles spiralling along Earth's magnetic field lines into the upper atmosphere.

The substance. The period does not depend on speed — faster particles move in bigger circles but take the same time per revolution, the principle behind the cyclotron.

**A straight conductor of length $l$ carrying current $I$ in a uniform field $B$ feels $\vec{F}=I\vec{l}\times \vec{B}$, of magnitude $IlB\sin \theta$, because the magnetic forces on all its drifting electrons add up.

Derivation.** The conductor holds $nAl$ free electrons, each drifting at $v_d$ and feeling $e{v}_{d}B$:

$$F=(nAl)(e{v}_{d}B)=(neA{v}_d)lB=IlB$$

In vector form, $\vec{F}=I\vec{l}\times \vec{B}$, with $\vec{l}$ along the current; Fleming's left-hand rule gives the direction.

Worked example. A $0.2$ m wire carries $5$ A at $30^{\circ }$ to a $0.4$ T field:

$$F=5\times 0.2\times 0.4\times \sin 30^{\circ }=0.2\text{ N}$$

Balancing weight. A horizontal wire can hover in a horizontal field when $IlB=mg$. For $m=10$ g, $l=0.5$ m and $B=0.2$ T:

$$I=\frac{mg}{lB}=\frac{0.01\times 9.8}{0.5\times 0.2}=0.98\text{ A}$$

An everyday example. A loudspeaker cone moves in and out because the varying current in its coil feels a changing force in the field of a permanent magnet.

The substance. A wire lying parallel to the field feels no force, since $\sin 0^{\circ }=0$.

**The Biot-Savart law gives the field of a small current element as $dB=\frac{{\mu }_0}{4\pi }\frac{Idl\sin \theta }{r^2}$, and adding these around a circular loop of radius $R$ gives $B=\frac{{\mu }_{0}I{R}^2}{2(x^2+R^2{)}^{3/2}}$ on its axis and $B=\frac{{\mu }_{0}I}{2R}$ at its centre.

The law in vector form:**

$$d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{Id\vec{l}\times \hat{r}}{r^2},{\mu }_0=4\pi \times 10^{-7}{\text{ T m A}}^{-1}$$

Circular loop, on its axis. Every element is $\sqrt{x^2+R^2}$ from a point on the axis. The components perpendicular to the axis cancel in pairs and the axial components add:

$$B=\frac{{\mu }_{0}I{R}^2}{2(x^2+R^2{)}^{3/2}}$$

At the centre, $x=0$, so $B=\frac{{\mu }_{0}I}{2R}$; for $N$ turns, multiply by $N$.

Worked example. A coil of $100$ turns and radius $0.1$ m carries $1$ A. At its centre:

$$B=\frac{N{\mu }_{0}I}{2R}=\frac{100\times 4\pi \times 10^{-7}\times 1}{2\times 0.1}=2\pi \times 10^{-4}\approx 6.28\times 10^{-4}\text{ T}$$

An everyday example. The coil inside an electric bell produces a concentrated magnetic field at its centre whenever current flows, just as this result predicts.

The substance. The Biot-Savart law is the magnetic partner of Coulomb's law — both fall as $\frac{1}{r^2}$, but the magnetic field is perpendicular to both the current element and $\hat{r}$.