How a Galvanometer Becomes Both an Ammeter and a Voltmeter

The needle of an analogue meter swings because a coil carrying current twists in a magnetic field. To understand it, we need the field of a long wire and a solenoid, the force between currents, and the torque on a loop.

This part covers Ampere's circuital law, the solenoid, parallel currents and the ampere, and the torque on a loop with the moving coil galvanometer.

**Ampere's circuital law states that the line integral of the magnetic field around any closed loop equals ${\mu }_0$ times the current passing through the loop, $\oint \vec{B}\cdot d\vec{l}={\mu }_{0}I$; for a long straight wire, a circular loop of radius $r$ gives $B=\frac{{\mu }_{0}I}{2\pi r}$.

Applying it to a long wire.** By symmetry, $B$ has the same size at every point of a circle of radius $r$ centred on the wire and points along the circle:

$$B\times 2\pi r={\mu }_{0}I\Rightarrow B=\frac{{\mu }_{0}I}{2\pi r}$$

Direction. By the right-hand thumb rule, the thumb points along the current and the curled fingers show the field.

Worked example. A wire carries $10$ A. At $5$ cm from it:

$$B=\frac{4\pi \times 10^{-7}\times 10}{2\pi \times 0.05}=\frac{2\times 10^{-6}}{0.05}=4\times 10^{-5}\text{ T}$$

An everyday example. Keeping a compass away from the thick cables of a home inverter avoids false readings caused by the fields around their currents.

The substance. **Ampere's law, like Gauss's law, always holds but is easy to use only when symmetry makes $B$ constant along the loop.**

**Inside a long solenoid the field is nearly uniform and along its axis, with magnitude $B={\mu }_{0}nI$, where $n$ is the number of turns per unit length, while the field outside is very weak.

Qualitative picture:

- The fields of neighbouring turns add inside the coil and largely cancel outside
- Field lines inside are straight and parallel, like those inside a bar magnet
- Near the ends, the field spreads out and weakens to about half its central value

Using the result:**

$$B={\mu }_{0}nI,n=\frac{N}{L}$$

Worked example. A solenoid $0.5$ m long has $500$ turns and carries $2$ A:

$$n=\frac{500}{0.5}=1000{\text{ m}}^{-1},B=4\pi \times 10^{-7}\times 1000\times 2\approx 2.5\times 10^{-3}\text{ T}$$

An everyday example. The electromagnet in an electric bell or a scrap-yard crane is a solenoid wound on iron, switched on and off by controlling the current.

The substance. The field depends on turns per unit length, not on total length — doubling both the turns and the length leaves $B$ unchanged.

**Each wire lies in the magnetic field of the other, so parallel wires carrying $I_1$ and $I_2$ a distance $d$ apart feel a force per unit length $\frac{F}{l}=\frac{{\mu }_0{I}_1{I}_2}{2\pi d}$ — attractive for currents in the same direction and repulsive for opposite currents — and this force defines the ampere.

Derivation:**

- Wire 1 produces $B_1=\frac{{\mu }_0{I}_1}{2\pi d}$ at wire 2
- A length $l$ of wire 2 feels $F=I_{2}l{B}_1=\frac{{\mu }_0{I}_1{I}_{2}l}{2\pi d}$

Worked example. Two wires $0.1$ m apart carry $5$ A and $10$ A in the same direction:

$$\frac{F}{l}=\frac{4\pi \times 10^{-7}\times 5\times 10}{2\pi \times 0.1}=1\times 10^{-4}{\text{ N m}}^{-1}$$

The force is attractive.

Definition of the ampere. One ampere is the steady current that, flowing in each of two very long, straight, parallel wires placed $1$ m apart in vacuum, produces a force of $2\times 10^{-7}$ N on each metre of wire.

An everyday example. The live and neutral wires in a household cable carry opposite currents, so they repel each other slightly — a force far too small to notice at normal currents.

The substance. Like currents attract, while like charges repel — the magnetic rule is the reverse of the electrostatic one.

**A coil of $N$ turns and area $A$ carrying current $I$ in a uniform field $B$ feels a torque $\tau =NIAB\sin \theta$, so it acts as a magnetic dipole with moment $m=NIA$; in a moving coil galvanometer a spring balances this torque, giving a deflection proportional to current, and a shunt or a series resistor converts it into an ammeter or voltmeter.

Torque on a loop:**

$$\vec{\tau }=\vec{m}\times \vec{B},\tau =NIAB\sin \theta$$

Moving coil galvanometer:

- A radial magnetic field keeps $\sin \theta =1$, so $\tau =NIAB$
- A spring provides a restoring torque $k\varphi$; at equilibrium $NIAB=k\varphi$
- Current sensitivity $\frac{\varphi }{I}=\frac{NAB}{k}$; voltage sensitivity $\frac{\varphi }{V}=\frac{NAB}{kG}$

Conversions:

- Ammeter — a small shunt in parallel: $S=\frac{I_{g}G}{I-I_g}$
- Voltmeter — a large resistance in series: $R=\frac{V}{I_g}-G$

Worked example. A galvanometer with $G=50\Omega$ gives full-scale deflection at $I_g=2$ mA.

$$\text{Ammeter up to }1\text{ A}:S=\frac{0.002\times 50}{1-0.002}\approx 0.1\Omega$$

$$\text{Voltmeter up to }10\text{ V}:R=\frac{10}{0.002}-50=4950\Omega$$

An everyday example. The needle of an electrician's analogue multimeter swings because a coil twists in a magnet's field, exactly as described here.

The substance. An ideal ammeter has almost zero resistance and an ideal voltmeter almost infinite resistance, so neither disturbs the circuit it measures.