Moving coil galvanometer

Slide a tiny direct current into a coil of wire hung between the poles of a magnet, and the coil twists — a little, a lot, always in proportion to the current. That twist, read off a pointer against a scale, is the whole idea behind the moving-coil galvanometer, the workhorse instrument for detecting and measuring small currents in every physics laboratory. It is not a lucky trick: it is the current-loop torque you already derived for a coil in a magnetic field, tamed into a linear, repeatable, readable device by one clever design choice — a magnetic field shaped so it is always radial to the coil. This article rebuilds the instrument piece by piece: why the radial field forces $I\propto\phi$ , how current sensitivity and voltage sensitivity are defined and why they can pull in different directions, and how to compute both from an instrument's coil data — reusing the exact numbers from the syllabus example, so nothing here is new physics, only a fuller look at material you have already met.

The core insight

A moving-coil galvanometer balances the magnetic deflecting torque $NBIA$ against a spring's restoring torque $k\phi$ , and a radial field keeps that balance linear, giving $I\propto\phi$ and two sensitivity measures, $\phi/I=NBA/k$ and $\phi/V=NBA/(kR_g)$ , that do not always improve together.

The core idea

Construction. A rectangular coil of $N$ turns of fine insulated wire is wound on a light aluminium frame and suspended (or pivoted) between the concave pole pieces of a strong permanent horseshoe magnet, with a soft-iron cylindrical core fixed inside the coil. A spring or torsion suspension supplies a restoring torque, and a pointer fixed to the coil sweeps a calibrated scale as the coil turns.

Why the field must be radial. The core and the curved pole pieces together shape the field so that $\vec B$ always lies in the plane of the coil, whatever angle the coil has turned through. That means the angle between $\vec B$ and the coil's normal is always $90°$ , so $\sin\theta=1$ at every instant of the swing, not just at the start. Without this shaping, $\theta$ would change as the coil rotated and the torque would vary non-linearly with angle, ruining the even scale.

Torque balance. With the field radial, the deflecting torque on the current-carrying coil is

\tau_{\text{deflecting}} = NBIA

where: $N$ is the number of turns, $B$ the radial field strength (T), $I$ the current (A), and $A$ the area enclosed by one turn (m $^2$ ). The spring resists with a torque proportional to the twist angle $\phi$ ,

\tau_{\text{restoring}} = k\phi

where: $k$ is the torsional constant of the suspension (N·m/rad). At the steady (equilibrium) deflection the two torques are equal, giving the operating relation $I=(k/NBA)\phi$ — current directly proportional to deflection, hence a linear, evenly spaced scale.

Figure

Torque balance diagram showing a rectangular coil between magnet poles with the magnetic torque NBIA rotating it one way and the spring torque kphi restoring it the other way until they balance

Derivation AFrom torque balance to the sensitivity formulas

Starting from the equilibrium condition that the magnetic deflecting torque on a current-carrying coil in a radial field exactly cancels the spring's restoring torque at the observed steady deflection, derive the working relation between current and deflection and then the two standard sensitivity measures used to characterise any moving-coil galvanometer, current sensitivity and voltage sensitivity.

Step 1. At equilibrium the deflecting torque equals the restoring torque: $NBIA = k\phi$ . This holds because the coil stops turning exactly when the two torques balance; nothing else acts on it once it settles (friction and air damping only affect how it settles, not where).

Step 2. Solve for $I$ in terms of $\phi$ : $I = (k/NBA)\,\phi$ . Because $N$ , $B$ , $A$ , and $k$ are fixed properties of a given instrument, this says current is directly proportional to deflection, $I\propto\phi$ , which the syllabus also writes as $I=k\phi$ using $k$ loosely for the combined instrument constant $k/(NBA)$ .

Step 3. Define current sensitivity as deflection produced per unit current, the reciprocal ratio: $\phi/I = NBA/k$ . A larger $N$ , $B$ , or $A$ , or a softer (smaller $k$ ) suspension, all raise current sensitivity — the coil swings further for the same current.

Step 4. Define voltage sensitivity as deflection per unit voltage applied across the coil terminals, using $V=IR_g$ where $R_g$ is the coil's own resistance: $\phi/V = \phi/(IR_g) = NBA/(kR_g)$ . This is current sensitivity divided by $R_g$ , so a coil with high current sensitivity but also high $R_g$ (from the extra turns needed to raise $N$ ) need not have correspondingly high voltage sensitivity — the two measures can move in opposite directions as the design changes.

Worked example

A moving-coil galvanometer has $N=100$ turns, each of area $A=2\times10^{-4}\ \text{m}^2$ , in a radial field $B=0.25\ \text{T}$ , with suspension constant $k=2\times10^{-6}\ \text{N}\,\cdot\,\text{m/rad}$ and coil resistance $R_g=50\ \Omega$ . Find its current sensitivity and its voltage sensitivity.

Step 1. Current sensitivity: $\phi/I = NBA/k = (100\times0.25\times2\times10^{-4})/(2\times10^{-6})$ .

Step 2. Numerator: $100\times0.25=25$ , and $25\times2\times10^{-4}=5\times10^{-3}$ . Dividing, $\phi/I = 5\times10^{-3}/2\times10^{-6} = 2500\ \text{rad/A}$ .

Step 3. Voltage sensitivity uses the current sensitivity just found, divided by $R_g$ : $\phi/V = (\phi/I)/R_g = 2500/50$ .

Step 4. $\phi/V = 50\ \text{rad/V}$ .

The current sensitivity is 2500 rad/A and the voltage sensitivity is 50 rad/V — despite the high current sensitivity, the $50\ \Omega$ coil resistance keeps voltage sensitivity comparatively modest, illustrating that the two are genuinely different design targets.

Competitive stretch

If the same coil were rewound with $N'=150$ turns (same wire gauge, so $R_g$ scales roughly with $N$ , giving $R_g'\approx75\ \Omega$ ), current sensitivity rises to $\phi/I = 1.5\times2500=3750\ \text{rad/A}$ , but voltage sensitivity becomes $3750/75=50\ \text{rad/V}$ — unchanged. This is the general pattern: adding turns of the same wire raises current sensitivity but leaves voltage sensitivity roughly fixed, because $R_g$ grows in step with $N$ .

Common traps

Thinking the radial field is a minor construction detail — it is what makes $\sin\theta=1$ hold at every deflection angle, and without it the scale would be cramped and non-linear rather than evenly spaced.
Mixing up current sensitivity and voltage sensitivity: they have different units (rad/A vs rad/V) and different dependence on $R_g$ , so a "sensitive" galvanometer in one sense need not be sensitive in the other.
Assuming more turns always makes a better instrument — more turns raises current sensitivity but also raises $R_g$ , so voltage sensitivity can stay flat or even fall.
Treating the bare galvanometer as ready to measure ordinary circuit currents — its coil can safely carry only a very small current before overheating or the pointer leaving the scale, which is exactly why conversion into an ammeter or voltmeter (using shunts and series resistances) is the next topic.

Ask the tutor: Want to see how this galvanometer gets converted into an ammeter or voltmeter using shunt and series resistances, or work through why doubling N leaves voltage sensitivity unchanged step by step — ask the AI tutor.

Why does a radial field give sin theta equals 1 instead of theta changing as the coil turns

The curved pole pieces plus the cylindrical iron core are shaped so that field lines always point straight into the core, radially, from wherever the coil surface currently sits. Because the coil's plane always contains the local field direction, the angle between the field and the coil's normal stays fixed at 90 degrees throughout the swing, so $\sin\theta=1$ at every deflection, not only when the coil starts undeflected. That is what keeps the torque proportional to $I$ alone, giving an evenly spaced scale instead of a cramped one.

If high current sensitivity sounds good why would I ever want a galvanometer with lower R_g

Current sensitivity tells you how far the pointer swings for a given current, which matters when you are directly measuring tiny currents. But if you plan to use the galvanometer as the basis of a voltmeter, what matters is voltage sensitivity, $\phi/V=(\phi/I)/R_g$ , and a large $R_g$ divides that down even if current sensitivity is high. A lower $R_g$ for the same current sensitivity would give a better voltmeter base, which is why instrument designers report both numbers rather than one.

Can I just read a current directly off a moving-coil galvanometer in a lab circuit

Not safely for ordinary circuit currents. The fine coil wire and the delicate suspension can only carry a very small current before the coil overheats or the pointer is driven off the end of the scale, since the instrument is built for high sensitivity to small currents, not for handling amperes. To use it for larger, calibrated currents or voltages you convert it into an ammeter or voltmeter using a shunt or series resistance, which is the topic that follows this one.

Moving coil galvanometer

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads