Construction and the radial field
The galvanometer consists of a coil, with many turns, free to rotate about a fixed axis, in a uniform radial magnetic field. There is a cylindrical soft-iron core which not only makes the field radial but also increases the strength of the magnetic field. When a current flows through the coil, a torque acts on it. This is exactly the torque on a current loop derived earlier in the chapter — the galvanometer is simply that physics packaged into an instrument.
The field is shaped by two concave pole pieces of a permanent magnet, with the soft-iron cylinder mounted between them. The gap between the curved poles and the core is a thin annular shell, so the field lines run radially — pointing straight toward (or away from) the axis at every point where the coil sits. A pointer attached to the suspension reads the deflection $\varphi$ on a scale.
Coil in a radial field between concave pole pieces and a soft-iron core.
Because the curved poles and the central core keep the field radial, the plane of the coil is always parallel to B. The coil normal stays perpendicular to B at every angle, so $\sin\theta = 1$ throughout the swing.
The balance: deflection proportional to current
The torque on the $N$-turn coil of area $A$ carrying current $I$ in field $B$ is, from the torque-on-a-loop result,
$$\tau = NI\,AB$$Since the field is radial by design, we have taken $\sin\theta = 1$ in this expression for the torque. The magnetic torque $NIAB$ tends to rotate the coil. A spring $\mathrm{Sp}$ provides a counter torque $k\varphi$ that balances the magnetic torque $NIAB$, resulting in a steady angular deflection $\varphi$. In equilibrium
$$k\varphi = NI\,AB$$where $k$ is the torsional constant of the spring; i.e. the restoring torque per unit twist. The deflection $\varphi$ is indicated on the scale by a pointer attached to the spring. We have
$$\varphi = \left(\frac{NAB}{k}\right) I$$The quantity in brackets is a constant for a given galvanometer. Hence the deflection is directly proportional to the current, $\varphi \propto I$, and the scale is uniform (linear). This linearity is the entire payoff of the radial-field design.
The radial field is what makes φ ∝ I linear
A common confusion is to carry over the loop torque $\tau = NIAB\sin\theta$ and expect the deflection to depend on $\sin\varphi$. In a galvanometer the field is radial, so $\theta = 90^\circ$ at every position of the coil and $\sin\theta = 1$ always. The torque stays $NIAB$ regardless of how far the coil has turned.
If a question removes the radial field (a coil in a uniform field), $\sin\theta$ returns and the scale is no longer linear. Radial field ⇒ $\varphi \propto I$.
Current sensitivity and voltage sensitivity
The galvanometer can be used in a number of ways — first as a detector to check if a current is flowing, as in the Wheatstone's bridge arrangement. To compare instruments we define two sensitivities directly from $\varphi = (NAB/k)\,I$.
We define the current sensitivity of the galvanometer as the deflection per unit current,
$$\frac{\varphi}{I} = \frac{NAB}{k}$$For voltage, the deflection responds to the current $I = V/R$ that the applied voltage $V$ drives through the coil resistance $R$. We define the voltage sensitivity as the deflection per unit voltage,
$$\frac{\varphi}{V} = \frac{NAB}{k}\cdot\frac{1}{R} = \frac{NAB}{kR}$$The two sensitivities therefore differ only by the factor $R$, the resistance of the galvanometer coil.
| Quantity | Definition | Expression | Depends on |
|---|---|---|---|
| Current sensitivity | Deflection per unit current, $\varphi/I$ | NAB / k | $N$, $A$, $B$, $k$ |
| Voltage sensitivity | Deflection per unit voltage, $\varphi/V$ | NAB / (kR) | $N$, $A$, $B$, $k$, $R$ |
| Relation | Voltage sens. = current sens. ÷ resistance | (φ/V) = (φ/I) / R | coil resistance $R$ |
This relation, $\dfrac{\varphi}{V} = \dfrac{1}{R}\cdot\dfrac{\varphi}{I}$, is itself a favourite NEET shortcut: given any two of current sensitivity, voltage sensitivity and resistance, the third follows immediately. The 2018 PYQ below is built on exactly this.
The starting torque $\tau = NIAB$ comes straight from the loop result — revisit Torque on a Current Loop to see where the $\sin\theta = 1$ simplification originates.
Why increasing turns may not raise voltage sensitivity
A convenient way for the manufacturer to increase the current sensitivity is to increase the number of turns $N$. From $\varphi/I = NAB/k$, if $N \to 2N$ then the current sensitivity doubles. However, the resistance of the galvanometer is also likely to double, since it is proportional to the length of the wire. In the voltage sensitivity $NAB/(kR)$, both $N \to 2N$ and $R \to 2R$, so the voltage sensitivity
$$\frac{\varphi}{V} \to \frac{(2N)AB}{k(2R)} = \frac{NAB}{kR}$$remains unchanged. An interesting point to note is that increasing the current sensitivity may not necessarily increase the voltage sensitivity. So in general, the modification needed for conversion of a galvanometer to an ammeter will be different from what is needed for converting it into a voltmeter.
"Add turns to make a better voltmeter" — false
Adding turns reliably improves current sensitivity, and students wrongly extend this to voltage sensitivity. The extra wire raises $R$ in step with $N$, and the two cancel in $NAB/(kR)$.
More turns ⇒ current sensitivity ↑, but voltage sensitivity ≈ unchanged (because $R \propto N$).
Conversion of a galvanometer to an ammeter
The galvanometer cannot as such be used as an ammeter to measure the value of the current in a given circuit. This is for two reasons: (i) the galvanometer is a very sensitive device, giving a full-scale deflection for a current of the order of $\mu$A; and (ii) for measuring currents the galvanometer has to be connected in series, and as it has a large resistance, this will change the value of the current in the circuit.
To overcome these difficulties, one attaches a small resistance $r_s$, called shunt resistance, in parallel with the galvanometer coil, so that most of the current passes through the shunt. The resistance of this arrangement is
$$\frac{R_G\, r_s}{R_G + r_s} \;\simeq\; r_s \qquad \text{if } R_G \gg r_s$$If $r_s$ has a small value in relation to the resistance $R_c$ of the rest of the circuit, the effect of introducing the measuring instrument is also small and negligible. The scale is then calibrated and graduated to read off the current value directly.
Ammeter: galvanometer G with a small shunt $r_s$ in parallel.
Most of the current is diverted through the low-resistance shunt; the combined resistance $\approx r_s$ is tiny, so inserting the ammeter in series barely disturbs the circuit.
Ammeter = LOW resistance in PARALLEL; voltmeter = HIGH resistance in SERIES
The single most-swapped pair on this topic. An ammeter goes in series and so must have low resistance — achieved by a small shunt in parallel. A voltmeter goes in parallel and so must have high resistance — achieved by a large $R$ in series.
Shunt $r_s$ → parallel → ammeter. Series $R$ (large) → voltmeter. Connection of the added resistor is opposite to how the meter is wired into the circuit.
Conversion of a galvanometer to a voltmeter
The galvanometer can also be used as a voltmeter to measure the voltage across a given section of the circuit. For this it must be connected in parallel with that section of the circuit. Further, it must draw a very small current, otherwise the voltage measurement will disturb the original set up by an amount which is very large. Usually we like to keep the disturbance due to the measuring device below one per cent.
To ensure this, a large resistance $R$ is connected in series with the galvanometer. The resistance of the voltmeter is now
$$R_G + R \;\simeq\; R : \text{ large}$$The scale of the voltmeter is calibrated to read off the voltage value directly.
Voltmeter: galvanometer G with a large $R$ in series, placed in parallel across a section.
The large series $R$ keeps the voltmeter's current tiny, so connecting it in parallel changes the section voltage by under one per cent.
Ideal ammeter vs ideal voltmeter
Pushing each design to its limit defines the ideal meters. An ideal ammeter has zero resistance — inserting it in series then adds nothing, so the circuit current is exactly the value being measured. An ideal voltmeter has infinite resistance — connected in parallel it then draws no current, so the section voltage is read without any disturbance. Real meters approach but never reach these limits.
| Ammeter | Voltmeter | |
|---|---|---|
| Measures | Current | Voltage (PD) |
| Connected | In series | In parallel |
| Made from G by adding | Small shunt $r_s$ in parallel | Large $R$ in series |
| Net resistance | $\dfrac{R_G r_s}{R_G + r_s} \approx r_s$ (low) | $R_G + R \approx R$ (high) |
| Ideal value | Zero resistance | Infinite resistance |
| Effect on circuit | Negligible if $r_s \ll R_c$ | Disturbance kept below ~1% |
Moving Coil Galvanometer in one glance
- A multi-turn coil rotates in a radial field set by concave pole pieces + a soft-iron core; the core also strengthens the field.
- Radial field ⇒ $\sin\theta = 1$ always ⇒ torque $\tau = NIAB$ at every angle.
- Balance against the spring: $k\varphi = NIAB$, so $\varphi = (NAB/k)\,I$ and $\varphi \propto I$ (uniform scale). $k$ = torsional constant.
- Current sensitivity $= NAB/k$; voltage sensitivity $= NAB/(kR)$; the two differ by the coil resistance $R$.
- Doubling $N$ doubles current sensitivity but leaves voltage sensitivity unchanged because $R$ also doubles.
- Ammeter: small shunt $r_s$ in parallel (low net R, in series). Voltmeter: large $R$ in series (high net R, in parallel).
- Ideal ammeter → zero resistance; ideal voltmeter → infinite resistance.