Moving coil galvanometer

Every current-measuring instrument you have ever seen with a needle sweeping across a dial — the analogue ammeter on a lab bench, the old-style voltmeter in a circuit box — is, underneath its casing, the same beautifully simple machine: a small coil of wire sitting in a magnetic field, twisting on its axis in exact proportion to the current flowing through it. That machine is the moving coil galvanometer, and it is the payoff of the entire "Moving Charges and Magnetism" chapter. Everything you learned about the force on a current-carrying wire, the torque on a current loop, and the current loop behaving like a tiny magnetic dipole was quietly building toward this one device. The genius of the galvanometer is not just that a current makes a coil feel a twisting force — that part follows directly from $\vec\tau = \vec m\times\vec B$ . The genius is in the engineering trick that makes the deflection track the current in a straight line, so that a uniformly marked scale — evenly spaced numbers, no crowding at one end — gives you an honest reading. That trick is a specially shaped magnet producing a radial magnetic field, and a hair-thin suspension fibre or spring that pushes back against the twist exactly the way a stretched spring pushes back against a pull. This article walks through why the radial field matters, derives the working equation for the equilibrium deflection, defines current sensitivity, and then does the two things every galvanometer is actually built for in real circuits: turning it into an ammeter with a shunt, and into a voltmeter with a series resistance. By the end you should be able to look at a galvanometer's coil dimensions and torsional constant and predict exactly how far the needle will swing — and how to bend that same instrument into a device that reads amperes or volts on demand.

Key takeaways

A moving coil galvanometer is a rectangular current-carrying coil suspended in a magnetic field; the magnetic torque on it twists a pointer, and the deflection is read off a scale.
A specially curved permanent magnet plus a soft-iron core inside the coil produces a radial magnetic field, keeping the field always in the coil's plane so the deflecting torque is $\tau_{\text{def}} = NIAB$ with no $\sin\theta$ factor.
The suspension (spring or fibre) supplies a restoring torque $\tau_{\text{res}} = k\phi$ that grows with the twist angle $\phi$ .
At equilibrium the two torques balance, giving $\phi = \dfrac{NAB}{k}I$ — deflection directly proportional to current, the property that makes the scale linear.
Current sensitivity $S_I = \phi/I = NAB/k$ measures how far the needle swings per unit current; it is raised by increasing $N$ , $A$ or $B$ , or lowering $k$ .
A shunt (low resistance in parallel) converts the galvanometer into an ammeter; a high resistance in series converts it into a voltmeter — the two conversions are mirror images of each other.
Key idea: because a radial magnetic field removes the sin θ factor from the torque, a moving coil galvanometer settles at a deflection φ = (NAB/k)I directly proportional to current, and adding a low-resistance shunt (parallel) or high resistance (series) turns that same coil into a working ammeter or voltmeter.

The core insight

A moving coil galvanometer balances a magnetic deflecting torque $NIAB$ against a spring's restoring torque $k\phi$ to give a deflection $\phi = (NAB/k)I$ proportional to current, and this single coil becomes an ammeter with a parallel shunt or a voltmeter with a series resistance.

Where it fits in the syllabus

The moving coil galvanometer is the explicitly listed capstone application of CBSE Class 12's Moving Charges and Magnetism chapter, sitting directly after torque on a current loop and the current loop as a magnetic dipole ( $m = NIA$ ). The board syllabus asks for the working principle and construction idea, current sensitivity, and the two standard conversions — to ammeter (shunt) and to voltmeter (series resistance) — as core, examinable content; a full quantitative derivation of the radial-field geometry itself is not demanded, only its consequence that $\sin\theta$ drops out. This section treats $\phi = (NAB/k)I$ as the one grounded formula worth deriving in full, and keeps the shunt/series-resistance results at the quoted, applied level the syllabus expects.

The physical picture

Picture a small rectangular coil of $N$ turns of insulated copper wire, wound on a light aluminium frame so it stays rigid but nearly massless, and suspended between the concave pole faces of a strong permanent magnet. A fixed cylindrical core of soft iron sits inside the coil, and the magnet's pole faces are curved to hug that core closely. This shape is not decorative — it forces the magnetic field lines to always point straight out along the radius wherever the coil happens to be, which is exactly what "radial field" means. Because the field is always radial, it is always in the plane of the coil (perpendicular to the coil's normal) no matter what angle the coil has rotated through — unlike a uniform field, where the torque depends on $\sin\theta$ and dies away as the coil rotates toward alignment. With a radial field that dependence is engineered away, and the torque stays at its maximum value for every deflection angle in the coil's working range.

\tau_{\text{def}} = NIAB

where: $N$ is the number of turns, $I$ is the current through the coil (amperes), $A$ is the area enclosed by one turn (square metres), and $B$ is the (radial) magnetic field strength at the coil (tesla). Physically, this says the twisting effect grows in direct proportion to the current — double the current and you double the torque, with no complications from the coil's orientation. The coil does not spin freely; it is held by a thin suspension fibre (or a pair of hairsprings in a pivoted design) that resists twisting the way a torsion spring resists a turn — the harder you twist it, the harder it pushes back, in direct proportion to the twist angle $\phi$ :

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

where: $k$ is the torsional constant of the suspension (newton-metre per radian), a property of the wire's stiffness and length, and $\phi$ is the angle the coil has rotated from its rest position. This is the rotational analogue of a stretched spring's restoring force $F = -kx$ .

Figure

A moving coil galvanometer schematic showing a rectangular coil between curved magnet pole faces with a soft iron core, radial field arrows pointing outward from the core through the coil at every position, and a pointer attached to the suspension deflecting across a scale

The coil settles wherever the deflecting torque and the restoring torque are equal in magnitude — the point where the spring has twisted just enough to stop the coil turning any further.

NIAB = k\phi \quad\Rightarrow\quad \phi = \dfrac{NAB}{k}\,I

where: every symbol is as defined above, and $\phi$ is the steady-state deflection you actually read off the scale. Since $N$ , $A$ , $B$ and $k$ are all fixed properties of a given instrument, this equation says $\phi \propto I$ — a straight-line relationship. That is precisely why a galvanometer's scale can be marked with evenly spaced divisions: twice the current gives exactly twice the deflection, all the way across the range. The ratio of deflection to current is called the current sensitivity:

S_I = \dfrac{\phi}{I} = \dfrac{NAB}{k}

where: $S_I$ has units of radians per ampere (or scale divisions per ampere in a practical instrument). A "more sensitive" galvanometer swings farther for the same tiny current — useful for detecting very small currents, but it also means the coil can carry only a very small maximum safe current $I_g$ before saturating the scale or overheating, which is exactly why real ammeters and voltmeters need the conversions below rather than using the bare galvanometer.

The torque balance derivation

Derivation ADeflection of a moving coil galvanometer in a radial field

The goal of this derivation is to obtain the working equation of the moving coil galvanometer, $\phi = (NAB/k)I$ , by balancing the magnetic torque that a current-carrying coil feels in a radial field against the mechanical restoring torque of its suspension, and to show why this balance — and only the radial-field geometry — produces a deflection that is a clean straight-line function of current rather than a curved, harder-to-read one.

Step 1. Start from the general torque on a current loop of magnetic moment $m = NIA$ in a uniform field $B$ : $\tau = NIAB\sin\theta$ , where $\theta$ is the angle between the coil's normal and $\vec B$ . In an ordinary uniform field this torque would fall to zero whenever the coil's plane lined up with the field, making the deflection–current relation nonlinear and the scale badly crowded near the ends.

Step 2. Bring in the radial-field construction: curved pole pieces plus a central soft-iron cylinder force the field to point along the radius everywhere the coil can sit, so the field is always parallel to the coil's plane (i.e. always at $90°$ to the coil's normal) for every rotation angle within the working range. Substituting $\theta = 90°$ so $\sin\theta = 1$ into the general torque expression, the deflecting torque simplifies to $\tau_{\text{def}} = NIAB$ , independent of $\phi$ .

Step 3. The suspension fibre or hairspring resists the twist with a restoring torque proportional to the angle turned, $\tau_{\text{res}} = k\phi$ , exactly as a torsion spring obeys Hooke's law in rotational form. As the current begins to twist the coil, $\phi$ grows from zero and $\tau_{\text{res}}$ grows with it, while $\tau_{\text{def}} = NIAB$ stays fixed for a given $I$ .

Step 4. The coil stops turning at the instant the two torques become equal — beyond that point the restoring torque would exceed the deflecting torque and pull the coil back; before it, the deflecting torque would win and push the coil further. Setting $\tau_{\text{def}} = \tau_{\text{res}}$ gives $NIAB = k\phi$ , and solving for $\phi$ gives the working equation $\phi = \dfrac{NAB}{k}I$ .

Step 5. Read the physics of the result. Because the right-hand side is a fixed instrument constant ( $NAB/k$ ) multiplying $I$ , the equilibrium deflection is directly proportional to the current with no trigonometric distortion — a straight line through the origin. This is precisely why the radial field is worth the extra manufacturing complexity: it converts a device that would otherwise have a curved, unreadable response into one with a uniform scale, and it is the single design choice that separates a moving coil galvanometer from a cruder tangent galvanometer.

Worked examples

Board level - current sensitivity and deflection of a lab galvanometer

A moving coil galvanometer has a coil of $N = 100$ turns, each of area $A = 2\times10^{-3}\ \text{m}^2$ , placed in a radial magnetic field of $B = 0.2\ \text{T}$ . The torsional constant of the suspension is $k = 1\times10^{-6}\ \text{N}\,\cdot\,\text{m/rad}$ . Find the current sensitivity, and the deflection produced by a current of $50\ \mu\text{A}$ .

Step 1. Write the current sensitivity formula: $S_I = \dfrac{NAB}{k}$ .

Step 2. Compute the numerator: $N A B = 100 \times (2\times10^{-3}) \times 0.2 = 4\times10^{-2}$ .

Step 3. Divide by $k$ : $S_I = \dfrac{4\times10^{-2}}{1\times10^{-6}} = 4\times10^{4}\ \text{rad/A}$ .

Step 4. Convert the given current to SI and multiply: $I = 50\ \mu\text{A} = 5\times10^{-5}\ \text{A}$ , so $\phi = S_I \times I = (4\times10^{4})(5\times10^{-5}) = 2\ \text{rad}$ .

The current sensitivity is $S_I = 4\times10^{4}\ \text{rad/A}$ , and a current of $50\ \mu\text{A}$ produces a steady deflection of $\phi = 2\ \text{rad}$ (about $115°$ — a large swing chosen here purely to keep the arithmetic clean; real instruments are designed to stay within a much smaller angular range).

Competitive level - converting the same galvanometer to a 2 A ammeter and a 10 V voltmeter

A galvanometer has coil resistance $G = 50\ \Omega$ and gives full-scale deflection for a current $I_g = 2\ \text{mA}$ . (a) Find the shunt resistance needed so the combination reads currents up to $I = 2\ \text{A}$ full-scale. (b) Find the series resistance needed so the combination reads potential differences up to $V = 10\ \text{V}$ full-scale. (c) Find the effective resistance of the ammeter combination and compare it with $G$ .

Step 1. For the ammeter, the shunt $S$ is placed in parallel with the galvanometer so that at full-scale, current $I_g$ flows through the galvanometer and the remaining $I - I_g$ flows through the shunt, and the two branches share the same voltage: $I_g G = (I - I_g)S$ , giving $S = \dfrac{I_g G}{I - I_g}$ .

Step 2. Substitute the numbers: $S = \dfrac{(2\times10^{-3})(50)}{2 - 0.002} = \dfrac{0.1}{1.998} \approx 5.0\times10^{-2}\ \Omega$ — a very small resistance, as expected, since it must carry almost the entire $2\ \text{A}$ while dropping the same tiny voltage the galvanometer branch drops.

Step 3. For the voltmeter, the series resistance $R$ must be large enough that at the maximum voltage $V$ , only $I_g$ flows through the whole series combination: $V = I_g(G + R)$ , so $R = \dfrac{V}{I_g} - G$ .

Step 4. Substitute: $R = \dfrac{10}{2\times10^{-3}} - 50 = 5000 - 50 = 4950\ \Omega$ — a large resistance, so the voltmeter draws only a tiny current even across $10\ \text{V}$ .

Step 5. The effective resistance of the ammeter (galvanometer parallel with shunt) is $R_A = \dfrac{GS}{G+S} = \dfrac{(50)(0.05)}{50.05} \approx 0.05\ \Omega$ , far smaller than $G = 50\ \Omega$ alone. Answers: $S \approx 0.05\ \Omega$ , $R = 4950\ \Omega$ , and the ammeter's effective resistance drops to about $0.05\ \Omega$ — this is exactly why a good ammeter must have very low resistance (it should not disturb the current it measures), while a good voltmeter must have very high resistance (it should draw negligible current from the circuit it measures). Timing note: about two minutes if the shunt and series-resistance formulas are memorised. Substitute values, keep $I_g$ in amperes throughout (a common slip is leaving it in milliamperes when subtracting from $I$ ), and always sanity-check the direction: shunt should come out tiny, series resistance should come out large — if either comes out the "wrong size" relative to $G$ , a sign or unit error has crept in.

Common traps

Board answers most often lose marks on the reasoning, not the arithmetic, in this topic. The most common slip is forgetting why the radial field matters at all — many students quote $\phi = (NAB/k)I$ correctly but cannot explain that it is the radial field which removes the $\sin\theta$ dependence and makes the scale linear; without that field, the general torque $NIAB\sin\theta$ would give an uneven, compressed scale. A second frequent error is mixing up the two conversions: the shunt for an ammeter is connected in parallel and is a low resistance, while the series resistance for a voltmeter is connected in series and is a high resistance — swapping either the placement or the relative size is an automatic mark loss. A third trap is treating $I_g$ carelessly in unit conversions — leaving it in milliamperes while every other quantity is in amperes (or vice versa) silently wrecks the shunt or series-resistance calculation, as the worked example above shows. Finally, students sometimes assume a bare galvanometer can be wired directly into a circuit as an ammeter or voltmeter; in fact its coil resistance and tiny maximum safe current $I_g$ mean it must always be modified with a shunt or series resistor before it can safely measure realistic currents or voltages.

Common trapMore sensitive is not automatically better

It is tempting to think that increasing current sensitivity $S_I = NAB/k$ (by adding turns, enlarging the coil area, strengthening the magnet, or using a softer suspension) is a pure improvement. It is not. A highly sensitive galvanometer swings to full scale for a very small current, which is exactly what makes it useless for measuring larger currents directly — it would saturate almost instantly. That is precisely why practical ammeters and voltmeters are built by taking a sensitive galvanometer and deliberately diverting most of the current around it (shunt) or dropping most of the voltage across an extra resistor (series resistance), rather than by trying to build a "less sensitive" galvanometer from scratch. Sensitivity and measuring range trade off against each other; the shunt and series-resistance conversions are how one instrument is adapted to serve very different ranges.

You should be able to…

Describe the construction of a moving coil galvanometer and explain why the radial magnetic field removes the $\sin\theta$ factor from the deflecting torque.
Derive the equilibrium deflection $\phi = (NAB/k)I$ from balancing $\tau_{\text{def}} = NIAB$ against $\tau_{\text{res}} = k\phi$ , and explain why this makes $\phi \propto I$ .
Define current sensitivity $S_I = NAB/k$ and state which quantities you would change, and how, to increase it.
Calculate the shunt resistance $S = I_gG/(I-I_g)$ needed to convert a galvanometer into an ammeter of a given range.
Calculate the series resistance $R = V/I_g - G$ needed to convert a galvanometer into a voltmeter of a given range.
Explain, in terms of not disturbing the measured circuit, why an ideal ammeter has very low resistance and an ideal voltmeter has very high resistance.

Ask the tutor: Want to work through why swapping the shunt and series-resistance formulas gives absurd answers, or set up a fresh numeric conversion problem with different target ranges for I and V?

Why does the deflection stop growing instead of the coil just spinning around forever?

Once current starts flowing, the magnetic torque twists the coil, but that same twist stretches or torques the suspension fibre, which pushes back harder the more it is twisted. The coil accelerates only while the magnetic torque exceeds the restoring torque; the instant they become equal, there is no net torque left to speed the coil up further, so it settles at that angle. It is the same idea as a mass on a spring finding a new equilibrium position when you hang an extra weight on it — the spring stretches until its pull balances the weight, not forever.

If a very sensitive galvanometer is bad for measuring large currents, why build sensitive galvanometers at all?

Sensitivity and range serve different jobs. A highly sensitive galvanometer is exactly what you want when the current itself is tiny and hard to detect — for example, checking for the faint current at a bridge's balance point. For measuring an ordinary circuit's 2 A, you do not build a separate, "less sensitive" instrument; you take that same sensitive galvanometer and add a shunt so almost all of the 2 A bypasses the delicate coil, letting only the small safe current Ig through it. One well-made sensitive galvanometer, adapted with different shunts or series resistors, can therefore serve as an ammeter or voltmeter of almost any range.

Why is the ammeter's resistance supposed to be low and the voltmeter's supposed to be high?

An ammeter is wired in series in the circuit, so any resistance it adds directly reduces the actual current flowing — a good ammeter should barely disturb the circuit, hence very low resistance (which is exactly what the tiny shunt achieves, pulling the combined resistance below even the bare coil's G). A voltmeter is wired in parallel across a component, so any current it draws is current stolen from that component's branch — a good voltmeter should draw almost none, hence very high resistance (which is what the large series resistor achieves). Both design goals point in the same direction: measure without disturbing.

Is the radial field the same thing as the coil having a soft iron core, or are they two separate things?

They work together but are two different parts of the design. The curved, concave pole faces of the permanent magnet are shaped to make the field point radially wherever the coil sits — that alone is the source of the radial geometry. The cylindrical soft-iron core placed inside the coil concentrates and strengthens the field further (soft iron is easily magnetised and channels field lines through it), and its cylindrical shape also helps keep the narrow air gap between magnet and core uniform, which is what actually makes the field radial and nearly constant in magnitude at every point the coil can reach. So the curved poles create the radial direction, and the iron core helps make it strong and uniform.