The Moving-Coil Galvanometer

Somewhere in your school lab sits a small box with a needle that twitches when a millionth of an ampere passes through it. No screen, no chip, no battery — yet it resolves currents a thousand times smaller than anything your body can feel, using nothing but a coil, a magnet and a hair-thin spiral of phosphor bronze. That box, the moving-coil galvanometer, is the ancestor of every analog meter ever made and one of the most elegant force-balance instruments in physics. The idea is a duel between two torques. Current through a coil sitting in a magnetic field produces a magnetic turning effect that grows with the current; a tiny spring fights back with a restoring twist that grows with the angle. The needle settles exactly where the two torques tie, so the angle is the current — read off a scale. The genius is in one geometric trick: shaping the magnet's poles so the field is radial, which makes the scale perfectly uniform from end to end. Miss that trick and the same instrument would need a differently-stretched scale for every part of its range. Boards love this instrument because it welds together everything Chapter 4 has built — force on a conductor, torque on a loop, and equilibrium — into one examinable machine. CBSE and NEET name its sensitivity explicitly in the syllabus; JEE asks the sly version, where doubling the turns somehow fails to improve the meter. Master the torque balance here, and the next module — converting this delicate detector into rugged ammeters and voltmeters — becomes pure algebra.

Key takeaways

A moving-coil galvanometer detects small currents by balancing the magnetic torque on a current-carrying coil, $\tau = NIAB$ , against a spiral spring's restoring torque $\tau = k\phi$ .
At equilibrium $NIAB = k\phi$ , so the deflection is $\phi = \dfrac{NAB}{k}\,I$ — deflection is directly proportional to current.
The radial field (concave pole pieces + soft-iron core) keeps the coil's plane parallel to the field at every position, pinning the torque at its maximum value and making the scale linear.
Current sensitivity $S_I = \phi/I = NAB/k$ ; voltage sensitivity $S_V = \phi/V = NAB/kG$ , where $G$ is the coil's resistance.
Adding turns raises $N$ and raises $G$ — so current sensitivity climbs while voltage sensitivity may not budge. This tradeoff is a standing JEE/NEET question.
Key idea: the needle parks where magnetic torque equals spring torque, and radial geometry makes that parking angle exactly proportional to the current.

The core insight

A current-carrying coil in a radial magnetic field experiences a constant-per-ampere torque $NIAB$ that twists it until the spring's restoring torque $k\phi$ balances it — so the equilibrium deflection $\phi = (NAB/k)\,I$ reads the current on a linear scale.

Where this lives today

The moving-coil movement never really retired — it changed jobs. Analog panel meters still ride on Weston's design wherever a glance beats a number: audio VU meters, aircraft standby gauges, battery-charge needles on bench supplies. The physics also runs in reverse throughout your day: send a signal through a coil in a radial magnetic gap and the torque-per-ampere law becomes motion-per-ampere — that is precisely a loudspeaker's voice coil, and the "galvo scanners" that steer laser beams in light shows, laser engravers and eye-surgery rigs are literally high-speed galvanometers with mirrors where the pointer used to be, swinging thousands of times per second under equation (3)'s proportionality. And when you next tap a digital multimeter, remember what it displaced: the DMM won on convenience, but it still must solve the problem Weston solved — turning a feeble electrical quantity into a trustworthy reading — just with an analog-to-digital converter where the torque balance used to be.

The setup

Strip the instrument to its five working parts. A rectangular coil of $N$ turns of fine insulated copper, wound on a light aluminium frame, hangs between the poles of a permanent horseshoe magnet. The pole faces are ground concave, and a soft-iron cylinder sits fixed inside the coil (the coil rotates around it, the cylinder does not move). Two phosphor-bronze hairsprings, one above and one below, do double duty: they feed current into and out of the coil, and they supply the restoring torque. A featherweight pointer rides on the coil's spindle over a graduated scale.

Figure

Moving-coil galvanometer in cross-section — concave poles and iron core shape a radial field through the coil

Each part answers a design question. Why concave poles and an iron core? Together they funnel the field so that in the gap it points along radii of the cylinder — a radial field — which is the linearity trick we prove below. Why phosphor bronze? It springs back reliably without tiring and conducts well, so the same spirals carry the current. Why an aluminium frame? As the coil swings, small currents induced in the metal frame oppose the motion and kill oscillations, so the needle settles quickly instead of wobbling — a damping bonus you will meet properly in the induction chapter. The driving physics comes from the two modules before this one. Force on a Current-Carrying Conductor gives each vertical side of the coil a force $F = BIl$ ; the two sides carry current in opposite directions, so the forces form a couple. Current Loops as Magnetic Dipoles: Torque & Moment packages that couple into a single result: a coil of $N$ turns and face area $A$ in a field $B$ feels a torque of magnitude $NIAB\sin\theta$ , greatest when the coil's plane lies parallel to the field. Here we spend that result on a measuring instrument.

The torque balance

Send a current $I$ through the coil. In the radial gap, the coil's plane is parallel to the field lines whatever its rotation angle (the field turns with it — that is what radial means), so the torque sits permanently at its maximum value:

\tau_{\text{deflecting}} = NIAB \qquad (1)

where: $N$ — number of turns; $I$ — the current being measured; $A$ — face area of one turn; $B$ — field strength in the gap. Notice what is missing: no $\sin\theta$ . The radial geometry has frozen the angle factor at 1, so every ampere buys the same torque at every needle position. That single deletion is the whole instrument. The hairsprings answer with a restoring torque proportional to how far they have been twisted:

\tau_{\text{restoring}} = k\phi \qquad (2)

where: $\phi$ — the deflection angle from rest; $k$ — the torsional constant of the springs (newton-metres of torque per radian of twist; a school instrument's $k$ is around $10^{-6}\ \mathrm{N\,m/rad}$ , which is why it can feel microamps). The stiffer the spring, the smaller the swing for the same current. The needle accelerates, overshoots a little, is damped by the aluminium frame, and parks where the torques tie. Setting (1) equal to (2):

\phi = \frac{NAB}{k}\,I \qquad (3)

where: everything in the bracket is fixed by construction — so $\phi \propto I$ . Double the current, double the angle, everywhere on the scale. Equation (3) is why the scale of a moving-coil instrument has evenly spaced markings, and it is the equation every board question about this instrument ultimately rearranges.

Derivation AWhy a radial field makes the scale linear

We want to show that with ordinary flat poles the galvanometer's scale would be non-uniform, and that the radial field created by concave poles and the iron core repairs this, giving deflection exactly proportional to current. The strategy is to write the torque balance twice — once in a uniform field where the angle factor survives, once in a radial field where it is pinned — and compare what each implies about the scale. Watch the $\sin\theta$ term: the entire design question is who controls that factor.

Step 1. Recall the general torque law from Current Loops as Magnetic Dipoles: a coil feels $\tau = NIAB\sin\theta$ , where $\theta$ is the angle between the coil's normal and the field. Torque is maximal ( $\sin\theta = 1$ ) when the plane of the coil is parallel to $B$ , and zero when the coil's face squares up to the field.

Step 2. Try flat poles first, giving a uniform horizontal field. Let the coil rest with its plane parallel to $B$ ( $\theta = 90°$ ). When the needle has swung by $\phi$ , the normal has rotated to $\theta = 90° - \phi$ , so the balance reads $NIAB\cos\phi = k\phi$ .

Step 3. Solve for the current: $I = \dfrac{k}{NAB}\cdot\dfrac{\phi}{\cos\phi}$ . The current is not proportional to the angle. Near zero the scale is nearly even, but at $60°$ the factor $\phi/\cos\phi$ has grown to twice the small-angle trend — equal current steps produce increasingly stretched needle steps, and the printed scale would have to be cramped at one end. Why this matters practically: a non-uniform scale must be calibrated point by point against known currents, and reading between marks becomes guesswork exactly where the divisions crowd.

Step 4. Now shape the poles concave and insert the soft-iron cylinder. Field lines leave the north face, cross the narrow gap along radii, pass through the iron, and exit radially to the south face. At any rotation angle, each vertical side of the coil sits in the gap with the local field lying in the coil's plane — the coil's normal stays perpendicular to the local field, $\sin\theta = 1$ permanently.

Step 5. The balance becomes $NIAB = k\phi$ at every position, so $\phi = (NAB/k)\,I$ : angle strictly proportional to current, uniform divisions from zero to full scale, and one calibration fixes the whole scale. Verified: at $\phi = 60° = 1.047\ \mathrm{rad}$ , the uniform-field expression gives $\phi/\cos\phi = 1.047/0.5 = 2.094$ , twice the radial-field value of $1.047$ — the numerical gap between a stretched scale and a linear one; recomputed directly.

The two sensitivities — and the tradeoff examiners love

A galvanometer's quality is quoted as how much swing per unit input. Two inputs matter, so there are two figures of merit.

Current sensitivity is deflection per ampere. Read it straight off equation (3):

S_I = \frac{\phi}{I} = \frac{NAB}{k} \qquad (4)

where: the units are radians per ampere (or, on real instruments, divisions per microampere). Want a more sensitive detector? Wind more turns, enlarge the coil face, use a stronger magnet, or fit a weaker spring — each lever appears exactly once.

Voltage sensitivity is deflection per volt applied across the terminals. The coil itself has resistance $G$ (fine copper wire, typically tens to hundreds of ohms), so a voltage $V$ drives $I = V/G$ through it, and:

S_V = \frac{\phi}{V} = \frac{NAB}{kG} \qquad (5)

where: $G$ — the galvanometer's own resistance. The same instrument now carries a second, different figure of merit, and here is the trap: the two do not move together. Every design lever that raises $S_I$ must be checked against what it does to $G$ .

Go deeper · Derivation B · 1 minDoubling the turns and the sensitivity that refuses to improve

Derivation BDoubling the turns and the sensitivity that refuses to improve

We want to show that winding twice as many turns on the same coil doubles the current sensitivity but can leave the voltage sensitivity unchanged — the single most-asked "paradox" about this instrument. The strategy is bookkeeping: track how $N$ and $G$ change together when the extra turns are wound from the same wire, then push both through equations (4) and (5). The thing to watch is the resistance: it scales with the length of wire, which scales with $N$ .

Step 1. Wind $2N$ turns of the same gauge wire on the same frame. Each turn has the same perimeter, so the total wire length doubles, and resistance is proportional to length: $G \to 2G$ . (Resistivity and cross-section unchanged — Resistivity & Geometry's $R = \rho L/\mathcal{A}$ at work.)

Step 2. Current sensitivity: $S_I = NAB/k \to 2NAB/k$ . Doubled, as advertised. As a current detector the instrument genuinely improved: the same microamp now produces twice the swing.

Step 3. Voltage sensitivity: $S_V = NAB/kG \to \dfrac{2NAB}{k\,(2G)} = \dfrac{NAB}{kG}$ . Exactly unchanged. The extra turns doubled the torque per ampere, but they also doubled the resistance, which halved the amperes per volt. The two factors of 2 cancel to the digit. Why the cancellation is physical, not accidental: a volt applied to the terminals pays for current through a longer wire; the torque gained per turn is repaid as current lost to resistance. Only levers that act on torque without touching the wire — a stronger magnet $B$ , a softer spring $k$ — improve both sensitivities at once.

Step 4. The exam version runs the logic with unequal factors. Raise $N$ by 50% while (for a thinner wire, say) $G$ doubles: $S_I$ gains the factor $1.5$ , but $S_V$ gains $1.5/2 = 0.75$ — the voltage sensitivity drops by 25% even as the current sensitivity climbs. Both statements are true of one instrument at the same time. Verified: with $S_I = 2000\ \mathrm{rad/A}$ and $G = 40\ \Omega$ giving $S_V = 50\ \mathrm{rad/V}$ : after the change $S_I = 3000\ \mathrm{rad/A}$ , $G = 80\ \Omega$ , $S_V = 3000/80 = 37.5\ \mathrm{rad/V} = 0.75 \times 50$ — recomputed.

What each design lever does

change	current sensitivity $S_I$	voltage sensitivity $S_V$	side effects
double $N$ (same wire)	×2	unchanged ( $G$ doubles too)	heavier coil, slower response
double $B$ (stronger magnet)	×2	×2	none on $G$ — the clean upgrade
double $A$ (bigger face)	×2	nearly ×2	perimeter grows, so $G$ creeps up
halve $k$ (softer spring)	×2	×2	fragile, slow to settle
halve $G$ (thicker wire)	unchanged	×2	bulkier coil for the same $N$
remove the iron core	scale turns non-linear	scale turns non-linear	the radial geometry collapses

Read the table like an engineer: the magnet is the only free lunch, which is why the twentieth century's meter industry chased ever-stronger permanent magnets. And read the last row like an examiner: the core is not decoration — pull it out conceptually and equation (3) dies, taking the uniform scale with it.

Figure

What the user sees — the radial field buys evenly spaced scale markings; a uniform field would crowd them

Worked examples

Board level — the full audit of one instrument (CBSE/ISC pattern)

A galvanometer coil has $N = 60$ turns, each of area $A = 2.0\times10^{-4}\ \mathrm{m^2}$ , in a radial field $B = 0.20\ \mathrm{T}$ . The suspension's torsional constant is $k = 1.2\times10^{-6}\ \mathrm{N\,m\,rad^{-1}}$ and the coil resistance is $G = 40\ \Omega$ . Find (a) the current sensitivity, (b) the current for full-scale deflection at $\phi = 15°$ , (c) the voltage sensitivity.

Step 1. Current sensitivity from equation (4): $S_I = \dfrac{NAB}{k} = \dfrac{60 \times 2.0\times10^{-4} \times 0.20}{1.2\times10^{-6}} = \dfrac{2.4\times10^{-3}}{1.2\times10^{-6}}$ . So $\boxed{S_I = 2000\ \mathrm{rad/A}}$ .

Step 2. Full-scale deflection in radians (the springs speak radians, not degrees): $\phi = 15° = 15 \times \dfrac{\pi}{180} = 0.262\ \mathrm{rad}$ .

Step 3. Full-scale current from $\phi = S_I\,I$ : $I_g = \dfrac{0.262}{2000} = 1.31\times10^{-4}\ \mathrm{A}$ , i.e. $\boxed{I_g \approx 131\ \mathrm{\mu A}}$ . A tenth of a milliamp swings the needle across the whole scale — this is a detector, not a workhorse ammeter.

Step 4. Voltage sensitivity from equation (5): $S_V = \dfrac{S_I}{G} = \dfrac{2000}{40} = \boxed{50\ \mathrm{rad/V}}$ .

Step 5. Sanity check the balance at full scale: magnetic torque $NIAB = 60 \times 1.31\times10^{-4} \times 2.0\times10^{-4} \times 0.20 = 3.1\times10^{-7}\ \mathrm{N\,m}$ ; spring torque $k\phi = 1.2\times10^{-6} \times 0.262 = 3.1\times10^{-7}\ \mathrm{N\,m}$ . They tie, as they must.

Competitive level — two sensitivities moving apart (JEE-Main/NEET pattern)

The galvanometer above is rewound so that the number of turns increases by 50%, but the thinner wire needed makes the coil resistance double. By what percentage does each sensitivity change?

Step 1. Current sensitivity carries only $N$ of the changed quantities: $S_I \propto N$ , so $S_I \to 1.5 \times 2000 = 3000\ \mathrm{rad/A}$ — an increase of 50%.

Step 2. Voltage sensitivity carries the ratio $N/G$ : factor $= \dfrac{1.5}{2} = 0.75$ , so $S_V \to 0.75 \times 50 = 37.5\ \mathrm{rad/V}$ — a decrease of 25%.

Step 3. Write both conclusions explicitly; the standard MCQ offers "both increase by 50%" as the bait option. One instrument, one modification, two opposite verdicts — because $S_V$ pays the resistance bill and $S_I$ does not. Timing note: 90 seconds. The only work is two ratios; the danger is autopiloting $S_V \propto N$ . Train the reflex: any rewinding question, write $S_I \propto N$ and $S_V \propto N/G$ side by side before touching numbers.

Exam traps

Degrees fed into $k\phi$ . Torsional constants are quoted per radian. Convert the deflection ( $15° = 0.262\ \mathrm{rad}$ ) before balancing torques; leaving degrees in place inflates every answer by a factor of 57.3. This single slip produces most wrong answers on part (b)-type questions.

A stray $\sin\theta$ in the radial field. Writing $\tau = NIAB\sin\theta$ and then agonising over the angle wastes time and invites error: in a working moving-coil instrument the radial field pins $\sin\theta = 1$ at all deflections. Reserve the $\sin\theta$ form for coils in uniform fields — that is the neighbouring module's territory, and mixing the two geometries is a classic mark-loser.

The two sensitivities conflated. "Sensitivity increases" is an incomplete sentence in this chapter. CBSE's syllabus line names current sensitivity; the JEE/NEET favourite is precisely that $S_I$ can rise while $S_V$ falls. Always compute both when the coil is modified.

Treating the galvanometer as an ammeter. Full-scale is microamps and the coil resistance is tens of ohms; putting this instrument directly in a circuit's main line either wrecks the reading or the coil. The repair — shunts and multipliers — is the next module, Galvanometer → Ammeter/Voltmeter Conversion, and examiners expect you to know the instrument is a detector, not a meter, until converted.

Units of the answer. Current sensitivity in $\mathrm{rad\,A^{-1}}$ (or $\mathrm{div/\mu A}$ on real scales), voltage sensitivity in $\mathrm{rad\,V^{-1}}$ , torsional constant in $\mathrm{N\,m\,rad^{-1}}$ . Boards award the unit mark separately — collect it.

Constants: these problems need no $g$ and no $\mu_0$ ; everything is in the given data. Keep $\pi$ symbolic until the final division.

Origins · 1 minEdward Weston and how this was discovered

Edward Weston

1850–1936 · Newark, New Jersey — the man who made the needle trustworthy

Weston trained as a physician in England, jumped ship to chemistry and electroplating in New York, and became obsessed with a scandal of the 1880s electrical boom: no two "voltmeters" agreed with each other. The laboratory galvanometers of the day were exquisite but temperamental — suspended by silk fibres, levelled with spirit bubbles, useless outside a basement. Weston attacked every failure point at once: a coil pivoted on jewelled bearings instead of a hanging fibre, springs of his own phosphor-bronze alloy that neither crept nor fatigued, an aged magnet that held its strength, and the radial-field geometry that kept the scale honest. His 1888 portable instrument read the same today as yesterday, in a factory as in a lab — the first electrical meter a working engineer could simply trust. The Weston movement became the template for essentially every analog panel meter of the next century, and his cadmium standard cell held the world's definition of the volt until 1990. The moving-coil galvanometer in your school lab is, part for part, Weston's argument that precision belongs on the workbench, not just in the academy.

History

1820

— Schweigger coils wire around a compass needle: the "multiplier," first current detector by deflection

1858

— Thomson's mirror galvanometer reads transatlantic telegraph pulses by bouncing light off a tiny rotating mirror

1882

— Deprez and d'Arsonval hang a coil in a radial magnet gap: the moving-coil movement with a linear scale is born

1888

— Weston's pivoted, spring-restored, production-built movement turns the laboratory instrument into a rugged industrial meter

1936

— Multi-range multimeters put a switched Weston movement in every technician's toolbox

Going further

Texts. NCERT Physics Class 12, Ch. 4 (Moving Charges and Magnetism, §4.10) — the CBSE/NEET canonical treatment · ISC Nootan/Kumar-Mittal Ch. on magnetic effects for the ISC phrasing · H.C. Verma, Concepts of Physics II, Ch. 34 problems for JEE-grade drill.

Provenance. All numbers recomputed: $S_I = 60\times2.0\times10^{-4}\times0.20/1.2\times10^{-6} = 2000\ \mathrm{rad/A}$ ; $I_g = 0.262/2000 = 131\ \mathrm{\mu A}$ ; $S_V = 2000/40 = 50\ \mathrm{rad/V}$ ; torque tie at $3.1\times10^{-7}\ \mathrm{N\,m}$ both sides; rewinding factors $1.5$ and $0.75$ ; division conversion $4.4\ \mathrm{\mu A/div}$ .

Prerequisites. Current Loops as Magnetic Dipoles: Torque & Moment (the $\tau = NIAB\sin\theta$ engine) · Force on a Current-Carrying Conductor (the per-side force). Leads to. Galvanometer → Ammeter/Voltmeter Conversion (THE classic next step) · The DC Motor & Its Cousins (the same couple, unleashed) · Real Meters — Loading Effects & Reading Errors (what converted meters do to circuits).

Why does the coil stop rotating at all — why not spin like a motor?

Because something fights back harder the further it turns. The magnetic torque on the coil is constant per ampere (radial field), but the hairsprings' opposing twist grows with angle, $k\phi$ . At some angle the two tie, and that equilibrium is stable: swing past it and the spring wins, pulling back; fall short and the magnetic torque wins, pushing on. A motor is the same coil with the spring removed and a commutator added so the torque never meets a balance — that story is The DC Motor & Its Cousins. A galvanometer is a motor deliberately forbidden to run.

What does the aluminium frame actually do? It is not in any formula.

It settles the needle. A coil on near-frictionless bearings would overshoot the balance angle, swing back, overshoot again — you would wait seconds for a reading. As the metal frame sweeps through the field, small induced currents circulate in it, and by Lenz's-law logic (coming in the induction chapter) they oppose the motion that creates them: a braking torque that exists only while the coil moves. The needle glides to rest almost without oscillation — "dead-beat" damping. The frame never appears in $\phi = (NAB/k)I$ because it acts only on the way to equilibrium, never at it.

Is a larger deflection always a larger current, even near the ends of the scale?

On a healthy instrument, yes — that is exactly what the radial field buys. $\phi = (NAB/k)I$ holds at 5% of full scale and at 95%, so equal current increments move the needle by equal amounts everywhere. If a real meter reads non-linearly, something has physically drifted: the magnet weakened, the core shifted, or the springs deformed. This is different from many other instruments (hot-wire meters, for example) whose scales are non-uniform by physics — a favourite comparison question on ISC papers.

Why is full-scale current so tiny? Can I not just pass more current?

The instrument is built soft on purpose: a feeble spring ( $k \sim 10^{-6}\ \mathrm{N\,m/rad}$ ) so that microamp-scale torques produce visible swings. Push milliamps through and the needle slams past full scale against its stop; push more and the fine copper wire overheats. The delicacy is the sensitivity — you cannot have both a hair-trigger detector and a rugged high-current meter in the same bare coil. The way out is external: divert most of the current around the coil with a shunt, or block most of the voltage with a series multiplier — the conversion mathematics of the next module.

Where exactly does the galvanometer sit in a circuit — series or parallel?

As a detector, wherever the question is "is any current flowing here?" — most famously the bridge arm of a Wheatstone bridge and the jockey lead of a meter bridge, where only the null reading matters and the galvanometer's own resistance becomes irrelevant at balance. Never straight across a battery (its tiny full-scale rating forbids it), and never bare in a mains-scale line. Once converted, its descendants take the standard postures: ammeter in series, voltmeter in parallel — and what those insertions do to the circuit is the loading story of Real Meters — Loading Effects & Reading Errors.

Current sensitivity is sometimes given in divisions per milliamp — how do I convert?

Bridge through the scale geometry. If full scale is $0.262\ \mathrm{rad}$ ( $15°$ ) spread over, say, 30 divisions, one division is $0.262/30 = 8.7\times10^{-3}\ \mathrm{rad}$ . A sensitivity of $2000\ \mathrm{rad/A}$ then means one division per $8.7\times10^{-3}/2000 = 4.4\ \mathrm{\mu A}$ — about $0.23$ divisions per microamp. Boards mostly stay in rad/A, but practical questions (and real instrument datasheets) speak divisions; the conversion is just the scale's radians-per-division ratio, and quoting it shows the examiner you can drive the actual instrument.

Does the Earth's magnetic field disturb the reading?

Negligibly, and by design. The gap field of the permanent magnet is of order $0.1$ – $0.5\ \mathrm{T}$ ; Earth's field is about $5\times10^{-5}\ \mathrm{T}$ — ten thousand times weaker, and mostly shorted through the iron core and yoke anyway. Contrast this with the older tangent galvanometers, which used Earth's field as their reference and had to be aligned north–south before every use. Freeing the instrument from the planet's field — so it works in any orientation, on a ship, in a factory — was one of the quiet victories of the moving-coil design.