In short

A moving coil galvanometer is a sensitive current-detecting instrument that converts a small electric current into a proportional angular deflection of a pointer across a calibrated scale.

Construction. A rectangular coil of N turns of fine insulated copper wire, wound on a light aluminium former, is suspended between the shaped pole pieces of a strong permanent magnet. A soft-iron cylinder sits inside the coil without touching it. A hairspring at each end provides the restoring torque; a pointer attached to the coil moves across a graduated scale.

Principle. When current I flows through the coil, the magnetic field B of the permanent magnet exerts a torque

\tau_\text{mag} \;=\; NIAB

on the coil (independent of angle, because the radial field ensures the coil plane always sits parallel to the field). The hairspring supplies a restoring torque \tau_\text{spring} = k\,\phi where k is the spring constant and \phi the deflection angle. At equilibrium,

\boxed{\;NIAB \;=\; k\,\phi \quad\Longrightarrow\quad I \;=\; \left(\frac{k}{NAB}\right)\phi\;}

so the current is proportional to the deflection and the scale can be marked linearly.

Current sensitivity. S_{I} = \phi/I = NAB/k, in radians per ampere (or divisions per ampere). Typical laboratory moving-coil galvanometers have S_{I}\sim 10^{5} div/A, allowing detection of currents as small as 10^{-6} A (one microampere per division).

Voltage sensitivity. S_{V} = \phi/V = NAB/(kR), where R is the galvanometer's coil resistance. Increasing the number of turns raises current sensitivity but also raises R — so doubling N doubles both NAB and R, leaving voltage sensitivity unchanged. Voltage sensitivity is a property of the material and geometry, not of the winding density.

Ammeter is made from a galvanometer by connecting a low-resistance shunt in parallel; voltmeter by adding a high-resistance multiplier in series. Both conversions use the same basic galvanometer mechanism.

Dead-beat galvanometer. An aluminium former for the coil provides eddy-current damping so the coil settles to its final reading in one swing (under a second), rather than oscillating. Ballistic galvanometer is the opposite — a moment-of-inertia-dominated design whose first-swing amplitude measures the total charge that passed.

Walk into any CBSE class 12 physics practical lab in Mumbai, Chennai, Lucknow, or Pune, and you will find the same instrument sitting on every bench: a rectangular wooden box, perhaps fifteen centimetres on a side, with a black-dialled face, a pointer hovering over a graduated scale, two brass terminals on top, and a small levelling screw under each foot. Marked on the face: "M.C. Galvanometer, 10\ \muA per division, R = 40\ \Omega". Wire it into a Wheatstone bridge and watch the pointer — the instant you balance the bridge, the pointer drops dead-centre to zero. That instrument, the moving coil galvanometer, is what lets you balance a bridge in the first place. It is the laboratory's go-to current detector, and it is sensitive enough to see a microampere.

The galvanometer was invented by Johann Schweigger in 1820 and refined to its modern form by Jacques-Arsène d'Arsonval and Edward Weston in the 1880s — but you do not need to remember the names. What matters is the physics. A tiny current flowing through a coil in a magnetic field feels a torque; a hairspring fights back; the angle the coil settles at is proportional to the current. That is the whole story, and the entire design of the instrument is a series of clever engineering choices to make that proportionality as linear, as sensitive, as stable, and as fast-settling as possible.

This chapter derives the torque equation from \vec{F} = I\vec{L}\times\vec{B}, explains why the pole pieces are shaped into concave faces and why a soft-iron cylinder sits at the heart of the coil, computes current and voltage sensitivity, shows you how to convert a galvanometer into an ammeter (for measuring amperes) or a voltmeter (for measuring volts), and closes with the damping theory that distinguishes a dead-beat galvanometer from a ballistic one.

Construction — inside the black wooden box

Take the lid off a standard laboratory galvanometer and this is what you see:

  1. A strong horseshoe permanent magnet with shaped concave pole pieces, N on one side and S on the other.
  2. A soft-iron cylinder fixed coaxially between the pole pieces, not touching them, leaving a narrow radial air gap.
  3. A rectangular coil of several hundred turns of very fine insulated copper wire, wound on a light aluminium former. The coil is suspended (by a fine phosphor-bronze strip above and a hairspring below, or by two hairsprings in a pivoted design) in the annular air gap between the pole pieces and the soft-iron cylinder.
  4. A pointer — a thin aluminium needle — fixed to the coil assembly, swinging across a graduated arc on the dial.
  5. Two terminals (marked "+" and "−") wired to the two ends of the coil through the hairsprings.
Moving coil galvanometer — cross-sectionCross-section of a moving coil galvanometer, showing horseshoe magnet with north and south pole pieces cupped around a central soft iron cylinder. A rectangular coil sits in the annular gap, suspended by a hairspring at top and bottom. Radial magnetic field lines connect the pole pieces to the iron cylinder. A pointer extends upward from the coil over a scale. N S soft iron coil (N turns) hairspring pointer hairspring −scale +scale + −
Top-view cross-section of a moving coil galvanometer. The rectangular coil (red outline) sits in the annular gap between the concave poles of the horseshoe magnet and the central soft-iron cylinder. The shaped poles plus the iron cylinder force the magnetic field lines to be *radial* in the gap (red-soft arrows) — so the field is always perpendicular to the plane of the coil no matter the coil's orientation. The pointer above extends over the scale arc.

The whole assembly is compact. The coil hangs in a narrow gap (a millimetre or so wide) where the magnetic field is made — by the shaped poles and the central iron cylinder — to be radial: at every point in the gap, the field points from the N pole straight inward to the iron, and from the iron straight outward to the S pole. This is the single most important engineering feature, and the next section shows why.

Deriving the torque — why the scale is linear

Step 1. The force on one side of the coil

A single straight wire of length L carrying current I in a magnetic field B (perpendicular to the wire) experiences a force

F \;=\; BIL, \tag{1}

perpendicular to both the wire and the field. (This is the magnetic force law you met in the chapter on forces on current-carrying conductors.)

For the rectangular coil of the galvanometer: let the two longer sides have length L (these are the "active" sides — vertical, running perpendicular to the page in the figure above) and the two shorter sides have length b (the horizontal top and bottom). The two active sides sit in the radial magnetic field B; the two short sides sit in a region of weak, complicated field and can be ignored for the torque calculation.

Each active side carries the same current I, but in opposite directions along the length. The force on one side:

F \;=\; BIL.

Step 2. Two forces, one torque

The two opposite sides of the rectangle feel forces in opposite directions — one is pushed out of the page, the other into it. (Use the right-hand rule: if I runs up on the left side and down on the right side, and B runs from left to right, the left-side force is out of the page and the right-side force is into the page.) These two forces form a couple — two equal, opposite, non-collinear forces, producing pure torque with zero net force.

The torque of a couple is: (magnitude of one force) × (perpendicular distance between the two force lines of action). The distance between the two active sides is b, so

\tau_\text{one turn} \;=\; F\cdot b \;=\; BIL\cdot b \;=\; BI\cdot(Lb) \;=\; BI\,A, \tag{2}

where A = Lb is the area of the rectangular coil.

Step 3. The N-turn coil

There are N turns of wire in the coil, all carrying the same current I. Each turn contributes its own BIA torque, in the same direction. The total magnetic torque is

\boxed{\;\tau_\text{mag} \;=\; NIAB.\;} \tag{3}

Why: the N turns are in series electrically (same current), and each turn's couple adds up to make the total torque N times larger than for a single turn. Equation (3) is the master formula for the deflecting torque in a galvanometer. Note that \tau_\text{mag} is independent of the coil's angular orientation, because the radial field means the field always sits in the plane of the coil regardless of how far the coil has rotated. This is crucial.

Step 4. Why the radial field matters — a linear scale

If the field were uniform (pointing, say, always left-to-right in the lab frame), the torque on the coil would depend on the angle \phi between the coil's normal and the field direction: \tau = NIAB\cos\phi. As the coil deflected, the effective torque would decrease, and the relationship between current and deflection would be nonlinear — a crowded scale at large deflections, sparse at small ones.

The shaped pole pieces plus the soft-iron cylinder turn the uniform magnet field into a radial field in the coil's gap. Now the field direction rotates with the coil: at whatever angle the coil sits, the field is still perpendicular to the coil's plane at every active side. The torque stays at NIAB, independent of \phi. So when the restoring spring torque is k\phi (linear in angle), the equilibrium condition NIAB = k\phi gives a deflection \phi that is exactly proportional to the current. The scale can be marked linearly, with equal divisions for equal increments of current.

This single engineering choice — the radial field achieved by concave pole faces plus a soft-iron central cylinder — is what makes a galvanometer scale uniform. Without it, the scale would compress near the ends and the instrument would be much harder to calibrate. The radial field is the difference between a toy and an instrument.

Step 5. The restoring torque — the hairspring

When the coil rotates, a hairspring (a thin, flat spiral spring of phosphor bronze) winds up and exerts a restoring torque proportional to the angular displacement:

\tau_\text{spring} \;=\; k\,\phi, \tag{4}

where k is the torsional spring constant (units: N·m/rad). The restoring torque always opposes the deflection — it tries to return the coil to \phi = 0.

Step 6. Steady-state equilibrium

At equilibrium, the deflecting torque equals the restoring torque:

NIAB \;=\; k\,\phi.

Solve for the deflection:

\boxed{\;\phi \;=\; \frac{NAB}{k}\,I.\;} \tag{5}

Why: equation (5) is the galvanometer equation. The deflection is linearly proportional to the current, with a proportionality constant NAB/k that depends only on the instrument's geometry, magnetic field strength, and spring constant — all fixed at the factory. Mark the scale in amperes once, and it reads amperes forever (until the magnet weakens, which takes decades).

Explore the deflection yourself

The interactive figure below lets you drag a current slider from zero up to 100 µA and watch the galvanometer's coil deflect accordingly. The parameters are for a typical CBSE-lab instrument: N = 200, A = 2\times 10^{-4} m² (a 2 cm × 1 cm coil), B = 0.2 T (a decent ferrite horseshoe), k = 4\times 10^{-8} N·m/rad (a soft hairspring). With these numbers, full-scale (\phi = 90° ≈ 1.57 rad) corresponds to a current of about I = k\phi/(NAB) = 4\times 10^{-8}\times 1.57/(200\times 2\times 10^{-4}\times 0.2) \approx 1.57\times 10^{-5}\ \text{A}\approx 16\ \muA. The slider caps at 100 µA just to show that beyond full scale the pointer hits the mechanical stop.

Interactive: deflection versus current for a laboratory galvanometer A linear relationship between deflection in degrees and current in microamperes, clipped at 90 degrees (full scale). A draggable point on the current axis selects a current value and the deflection reads out. current I (µA) deflection φ (degrees) 0 20 40 60 80 100 0 30 60 90 φ = (NAB/k) × I full scale (mechanical stop) drag the red point along I
Drag the red point along the current axis. A current of 8 µA — roughly what you might source from a freshly balanced Wheatstone bridge's null point — produces a deflection of about 92° (off-scale), showing why you need a galvanometer with an even softer hairspring for bridge-null work. Beyond the full-scale of 90°, the pointer hits a mechanical stop; real galvanometers mark the scale only up to ~80% of full scale to leave head-room.

Current sensitivity and voltage sensitivity

Current sensitivity

The current sensitivity is the deflection per unit current:

\boxed{\;S_{I} \;=\; \frac{\phi}{I} \;=\; \frac{NAB}{k}.\;} \tag{6}

Units: radians per ampere, or (more commonly for laboratory instruments) divisions per microampere. A sensitive laboratory galvanometer can have S_{I}\sim 100 div/µA, meaning a 1 µA current produces a 100-division deflection — easily readable on the scale. Commercial tangent galvanometers used in practical physics labs typically have S_{I}\sim 1–10 div/µA.

How to increase current sensitivity:

Voltage sensitivity

If the galvanometer is connected across a voltage source V, the current flowing is I = V/R_{g} where R_{g} is the coil's own resistance. The deflection is then

\phi \;=\; S_{I}\,I \;=\; \frac{NAB}{k}\cdot\frac{V}{R_{g}}.

The voltage sensitivity is

\boxed{\;S_{V} \;=\; \frac{\phi}{V} \;=\; \frac{NAB}{k\,R_{g}}.\;} \tag{7}

Why: voltage sensitivity is current sensitivity divided by the coil resistance. It is the deflection per volt of applied voltage.

Why increasing N doesn't help voltage sensitivity

Suppose you double N by winding twice as many turns of the same wire. The wire length doubles, so the coil resistance R_{g} also doubles. In equation (7):

S_{V} \;=\; \frac{N\,AB}{k\,R_{g}}

both N (numerator) and R_{g} (denominator) double, so S_{V} stays the same.

This is one of the deepest little facts in instrument design: voltage sensitivity is a property of the magnet and the spring, not of the winding density. Once you have chosen the magnet and the spring, you cannot improve voltage sensitivity by winding more turns. You can improve current sensitivity, yes — but at the cost of higher coil resistance that cancels the gain when voltage is the input.

What does improve voltage sensitivity: using thicker wire (lower R per turn, but fewer turns fit in the same gap — net cancellation); using a stronger magnet or a softer spring; making the coil physically larger. Instrument makers optimise all three within the space-and-cost budget.

Converting a galvanometer into an ammeter or voltmeter

A galvanometer on its own measures at most microamperes. An ammeter reads amperes; a voltmeter reads tens of volts. Both are built on top of the galvanometer movement by adding external resistors.

Ammeter — add a shunt in parallel

To measure a large current I with a galvanometer that can handle only a small current I_{g} at full scale, bleed most of the current around the galvanometer through a parallel low-resistance shunt R_{s}.

Converting a galvanometer to an ammeter with a shuntA galvanometer with coil resistance R_g in parallel with a low resistance shunt R_s. Current I enters, splits into I_g through the galvanometer and (I - I_g) through the shunt, rejoins, and exits. G R_g I_g → R_s (shunt) I − I_g → I I
Shunting a galvanometer to read amperes. The galvanometer branch carries the small current $I_{g}$ at full scale; the shunt carries the remaining $I - I_{g}$. Both are in parallel, so both see the same voltage drop.

The two branches are in parallel, so the voltage across the galvanometer equals the voltage across the shunt:

I_{g}\,R_{g} \;=\; (I - I_{g})\,R_{s}.

Solve for R_{s}:

\boxed{\;R_{s} \;=\; \frac{I_{g}\,R_{g}}{I - I_{g}}\;} \tag{8}

Why: choose R_s so that when the total current is I (the desired full-scale reading), exactly I_g (the galvanometer's own full-scale current) flows through the coil and the rest goes around. If I \gg I_g (which is the usual case), R_s \approx I_g R_g / I — the shunt is tiny compared to the coil resistance.

Numerical example. A galvanometer with R_{g} = 40\ \Omega and full-scale current I_{g} = 10 µA is to be converted into an ammeter of 1 A full scale.

R_{s} \;=\; \frac{10^{-5}\times 40}{1 - 10^{-5}} \;\approx\; 4\times 10^{-4}\ \Omega.

A 0.4 milliohm shunt — a short stubby piece of low-resistance metal (manganin) bolted across the galvanometer's terminals. The resulting instrument has an effective resistance of R_{g}\parallel R_{s} \approx 4\times 10^{-4} Ω: very low, as an ideal ammeter should be.

Voltmeter — add a multiplier in series

To measure a large voltage V with a galvanometer that at full scale drops only I_{g}R_{g} across itself, put a high-resistance multiplier R_{m} in series.

Converting a galvanometer to a voltmeter with a multiplierA galvanometer with coil resistance R_g in series with a high resistance multiplier R_m, connected across a voltage V. The same current I_g flows through both. G R_g R_m + − V = I_g (R_g + R_m) I_g
Adding a series multiplier turns a galvanometer into a voltmeter. The multiplier drops most of the voltage; only $I_{g}R_{g}$ falls across the galvanometer itself.

The same current I_{g} flows through both R_{g} and R_{m} (they are in series). The total voltage across them is

V \;=\; I_{g}(R_{g} + R_{m}).

Solve for R_{m}:

\boxed{\;R_{m} \;=\; \frac{V}{I_{g}} - R_{g}\;} \tag{9}

Why: pick R_m so that when the full-scale voltage V is applied, the series current equals I_g — which sends the galvanometer to full scale.

Numerical example. The same 10 µA, 40 Ω galvanometer, to be converted into a 10 V full-scale voltmeter:

R_{m} \;=\; \frac{10}{10^{-5}} - 40 \;=\; 10^{6} - 40 \;\approx\; 10^{6}\ \Omega \;=\; 1\ \text{M}\Omega.

A 1-megaohm resistor in series. The voltmeter's total resistance is about 1 MΩ — very high, as an ideal voltmeter should be (drawing negligible current from the circuit under test).

Note the pleasing symmetry: the shunt is tiny (milliohms), the multiplier is huge (megaohms), both for the same underlying galvanometer. Ammeter = low impedance = parallel shunt; voltmeter = high impedance = series multiplier. The galvanometer is the common soul of both.

Worked examples

Example 1: Designing a ammeter from a lab galvanometer

A CBSE-kit moving coil galvanometer reads full scale at I_{g} = 50 µA and has a coil resistance R_{g} = 60\ \Omega. You want to convert it into an ammeter that reads 0–3 A at full scale for use in a demonstration of Joule heating. Find the shunt resistance required and the effective resistance of the resulting ammeter.

Ammeter conversion circuit for worked example 1Galvanometer G with 60 ohm coil resistance in parallel with a shunt R_s. Current 3 amps enters and leaves at the terminals, with 50 microamps through G and 2.99995 amps through R_s. G 60 Ω 50 µA R_s (shunt) ~3 A 3 A
The ammeter configuration for Example 1: galvanometer G in parallel with shunt $R_{s}$, reading 3 A at full scale.

Step 1. Identify knowns. I_{g} = 50\ \muA = 5\times 10^{-5} A, R_{g} = 60\ \Omega, I = 3 A (desired full scale).

Why: I_g is the full-scale current the galvanometer can safely take without the pointer banging into the stop. R_g is the coil's DC resistance, given on the instrument's nameplate.

Step 2. Apply the shunt formula (8).

R_{s} \;=\; \frac{I_{g}\,R_{g}}{I - I_{g}} \;=\; \frac{5\times 10^{-5}\times 60}{3 - 5\times 10^{-5}} \;=\; \frac{3\times 10^{-3}}{2.99995}.

Why: substitute directly. The denominator is essentially 3, because the galvanometer current is negligible compared to the shunt current.

Step 3. Evaluate numerically.

R_{s} \;=\; \frac{3\times 10^{-3}}{3} \;=\; 1.0\times 10^{-3}\ \Omega \;=\; 1\ \text{m}\Omega.

Why: a one-milliohm shunt — a short, thick strip of manganin (a copper-manganese-nickel alloy with very small temperature coefficient). You cannot fabricate this out of copper wire; you would need a geometry several centimetres long of a low-resistance alloy or a precision resistor of that value.

Step 4. Compute the effective ammeter resistance.

R_{A} \;=\; R_{g}\parallel R_{s} \;=\; \frac{R_{g}\,R_{s}}{R_{g}+R_{s}} \;=\; \frac{60\times 10^{-3}}{60.001} \;\approx\; 10^{-3}\ \Omega.

Why: parallel combination is dominated by the smaller resistance. Since R_s \ll R_g, R_A \approx R_s. An ammeter with 1 mΩ of internal resistance inserts negligible voltage drop into a circuit measuring amperes at volts of supply.

Result: R_{s} \approx 1 mΩ. The ammeter has effective resistance 1 mΩ and full-scale reading 3 A.

What this shows: The shunt is physically a beefy strip of low-temperature-coefficient alloy — not a wimpy resistor. In the lab it is usually bolted directly across the galvanometer terminals, sometimes built into the instrument case. Commercial multimeters use a rotary-switch bank of such shunts, one per current range (30 mA, 300 mA, 3 A, 30 A).

Example 2: Voltage sensitivity and a bridge-balance problem

A laboratory galvanometer has N = 250 turns of wire, coil area A = 3\ \text{cm}^{2} = 3\times 10^{-4}\ \text{m}^{2}, magnetic field B = 0.25 T, torsional spring constant k = 5\times 10^{-8} N·m/rad, and coil resistance R_{g} = 80\ \Omega. (a) Find its current sensitivity and voltage sensitivity. (b) A Wheatstone bridge connected to this galvanometer has a 2 mV imbalance across the galvanometer. What deflection does the pointer show?

Step 1. Compute the current sensitivity from (6).

S_{I} \;=\; \frac{NAB}{k} \;=\; \frac{250\times 3\times 10^{-4}\times 0.25}{5\times 10^{-8}}.

Numerator: 250\times 3\times 10^{-4}\times 0.25 = 0.01875 N·m/A (torque per amp).

S_{I} \;=\; \frac{0.01875}{5\times 10^{-8}} \;=\; 3.75\times 10^{5}\ \text{rad/A}.

Why: current sensitivity in radians per amp. Dividing by \pi/180 ≈ 0.01745 gives degrees per amp: about 2.15\times 10^{7} deg/A, or 21.5 deg/µA. A clearly readable deflection.

Step 2. Compute the voltage sensitivity from (7).

S_{V} \;=\; \frac{S_{I}}{R_{g}} \;=\; \frac{3.75\times 10^{5}}{80} \;=\; 4.69\times 10^{3}\ \text{rad/V}.

Why: voltage sensitivity = current sensitivity divided by coil resistance. In degrees per volt: 2.69\times 10^{5} deg/V, or 0.269 deg/mV.

Step 3. Compute the deflection for the 2 mV bridge imbalance.

\phi \;=\; S_{V}\,V \;=\; 4.69\times 10^{3}\times 2\times 10^{-3} \;=\; 9.38\ \text{rad}.

That is unphysical — it is larger than 2\pi. The pointer cannot deflect more than 90° from the zero position without hitting the mechanical stop. What the real galvanometer does: the pointer slams hard against the stop and stays there.

Why: the calculation says the galvanometer would deflect 537° if it could. It can't. It pegs at full scale. This tells you the bridge is massively unbalanced — probably one of the ratio arms has a resistor an order of magnitude off from the others. You need to add a series resistor (about 1 kΩ) between the bridge and the galvanometer to reduce its sensitivity and actually read something meaningful.

Step 4. Redesign the measurement. To bring the full-scale deflection down to a sensible 2 mV, add a series resistance R_{\text{series}} such that

\phi_\text{fs} \;=\; \frac{NAB}{k(R_{g} + R_\text{series})}\,V_\text{fs}.

Setting \phi_\text{fs} = \pi/2 (90°) and V_\text{fs} = 2 mV:

R_{g} + R_\text{series} \;=\; \frac{NAB\cdot V_\text{fs}}{k\cdot\phi_\text{fs}} \;=\; \frac{0.01875\times 2\times 10^{-3}}{5\times 10^{-8}\times 1.571} \;=\; \frac{3.75\times 10^{-5}}{7.85\times 10^{-8}} \;\approx\; 478\ \Omega.

So R_\text{series} = 478 - 80 = 398\ \Omega. A 390 Ω standard resistor placed in series puts the 2 mV full-scale imbalance at the 90° mark.

Why: the original galvanometer is too sensitive for this application. Inserting a series resistor of about 400 Ω reduces the effective sensitivity by a factor of about 478/80 \approx 6 and makes the instrument usable for the task. This is exactly the kind of design choice a practical experimenter makes.

Result: S_{I} = 3.75\times 10^{5} rad/A, S_{V} = 4.69\times 10^{3} rad/V. The 2 mV imbalance would peg the pointer; a 390 Ω series resistor brings it to full scale at 2 mV — the practical fix.

What this shows: Sensitivity is a two-edged sword. A galvanometer so sensitive that 2 mV pegs it is too sensitive for most applications. Engineers sacrifice some sensitivity by adding a known series resistor, trading deflection-per-volt for a sensible readable range. The same galvanometer, with the 390 Ω resistor, becomes a practical millivoltmeter.

Common confusions

If you came here to understand how a galvanometer works and how to convert it into an ammeter or voltmeter, you have it. The rest is for readers who want the damping theory that distinguishes dead-beat from ballistic galvanometers, and the formal statement of why the radial-field design matters.

The equation of motion and damping

The coil's moment of inertia is J, the restoring torque is k\phi, and there is a damping torque -b\,\dot\phi from eddy currents and air friction. Newton's second law for rotation:

J\,\ddot\phi \;+\; b\,\dot\phi \;+\; k\,\phi \;=\; NIAB. \tag{10}

This is the equation of a damped harmonic oscillator driven by the constant torque NIAB. The behaviour depends on the discriminant b^{2} - 4Jk:

  • Under-damped (b^{2} < 4Jk): the pointer oscillates about the equilibrium angle with a frequency \omega = \sqrt{k/J - (b/2J)^{2}}, decaying in amplitude. Oscillations may take tens of seconds to die down — unsuitable for routine reading.
  • Over-damped (b^{2} > 4Jk): the pointer creeps slowly to equilibrium with no oscillation, taking a long time to settle — also unsuitable.
  • Critically damped (b^{2} = 4Jk): the pointer reaches equilibrium in the shortest possible time without overshooting. This is the dead-beat condition, and it is the design target for routine laboratory galvanometers.

The aluminium former — eddy currents as perfect damping

Wind the coil on a solid aluminium former (an aluminium rectangle that the wire is wound around). As the coil rotates, the aluminium itself moves through the magnetic field. By Faraday's law, this induces eddy currents in the aluminium; by Lenz's law, these eddy currents produce a magnetic force that opposes the rotation. The damping is automatic, strong, and scales with the rotation speed — the definition of linear damping -b\dot\phi.

The aluminium former is chosen so that the eddy-current damping puts the system very close to critical damping. The pointer reaches its new equilibrium in about one swing, without oscillation. Open a CBSE-kit galvanometer and you will see the aluminium framework behind the coil — that bit of scrap aluminium is the galvanometer's secret to usability.

Ballistic galvanometer — the opposite design

If instead of steady current you want to measure the total charge q = \int I\,dt that passes during a brief pulse (a capacitor discharge, a flux change in a coil), use a ballistic galvanometer — designed with:

  • A heavy coil (large J) to resist rapid rotation.
  • Weak damping (b \to 0), achieved by winding the coil on a non-conducting silk or bakelite former (no eddy currents).

During the brief pulse, the impulsive torque \int NIAB\,dt = NAB\int I\,dt = NAB\cdot q gives the coil an angular impulse, not a deflection. The coil then swings out under nearly zero damping until the spring's stored energy equals the kinetic energy it was given. The first-swing amplitude \phi_\text{max} is therefore proportional to q:

\phi_\text{max} \;=\; \frac{NAB}{\sqrt{Jk}}\,q \;=\; \frac{NAB}{k}\,\omega_{0}\,q, \qquad \omega_0 = \sqrt{k/J}.

A ballistic galvanometer thus reads charge directly from the first-swing amplitude — useful for measuring the charge stored on a capacitor or the magnetic flux change through a search coil in a Rowland-ring measurement.

Why the radial field gives a rigorously linear scale — the formal statement

In a uniform magnetic field, the torque on a current loop is \vec{\tau} = \vec{m}\times\vec{B} where \vec{m} = NI\vec{A} is the magnetic moment. Magnitudes: \tau = NIAB\sin\theta where \theta is the angle between the loop normal and the field. For small angles, \sin\theta \approx \theta and the torque is proportional to \theta. For large angles, the small-angle approximation fails and the scale becomes nonlinear.

In a radial field, the component of \vec{B} perpendicular to the plane of each active side of the coil is B at every angle (the field rotates with the coil). The torque is then NIAB independently of the angle. The rigorous statement: the shaped pole pieces and the soft-iron central cylinder together implement a cylindrical-symmetric field such that the magnetic flux through the coil is independent of the coil's rotation angle (over the useful range of motion). This breaks the usual \sin\theta dependence and gives an exactly linear torque. This is the single most important design innovation in the moving-coil galvanometer and is what earned d'Arsonval and Weston their enduring reputation in instrument-making.

Modern replacements — the digital multimeter

A digital multimeter uses a high-impedance amplifier and an analog-to-digital converter instead of a mechanical coil-and-pointer. No moving parts; no hairspring drift; no magnet aging; a reading accurate to 0.01%. So why are analog galvanometers still around?

  • Intuitive null detection: when balancing a Wheatstone bridge, you want zero deflection. A needle that sits rock-still at the middle mark is immediately obvious; a digital display flickering between "−0.1 mV" and "+0.1 mV" is less intuitive.
  • Ballistic charge measurement: a digital meter cannot measure a brief charge pulse without a custom integrator circuit; a ballistic galvanometer does it with its coil's inertia.
  • Educational value: a CBSE practical examiner wants to see the student understand the physics of a current-carrying coil in a magnetic field. A digital multimeter hides that physics behind a microprocessor; a moving-coil galvanometer puts it on display.

For precision DC measurement, the digital multimeter has won. For intuition, teaching, and ballistic charge measurement, the galvanometer remains the instrument of choice.

Where this leads next