Moving Coil Galvanometer

When a conductor of length $L$ carrying current $I$ is placed in a magnetic field $B$, it experiences a force $F = BIL\sin\theta$. For a rectangular coil with $N$ turns, width $b$, and length $l$, both vertical sides (length $l$) experience forces. In a radial magnetic field, the plane of the coil is always parallel to the field lines, so $\sin\theta = 1$ at all positions, ensuring a linear deflection scale. The two forces on the vertical sides form a couple, producing a deflecting torque $\tau_d = NBIA$, where $A = l \times b$ is the area of the coil.

The galvanometer consists of: (1) A strong permanent horseshoe magnet with concave pole pieces to produce a radial magnetic field. (2) A soft iron cylindrical core placed inside the coil — this not only concentrates the field but ensures it remains radial as the coil rotates. (3) A rectangular coil of fine insulated copper wire wound on a lightweight aluminium frame, pivoted between the poles. (4) Two phosphor-bronze spiral springs (upper and lower) that provide the restoring torque and also serve as current leads into the coil. (5) A lightweight aluminium pointer attached to the coil that moves over a calibrated scale. The aluminium frame also provides electromagnetic damping: as the coil rotates, eddy currents induced in the frame create a braking force that prevents oscillation.

When current $I$ flows through the coil, the deflecting torque is $\tau_d = NBIA$. The spiral springs resist this rotation with a restoring torque $\tau_r = k\phi$, where $k$ is the spring constant (torsional stiffness in N·m/rad) and $\phi$ is the angle of deflection. At equilibrium, $\tau_d = \tau_r$, giving: $$NBIA = k\phi$$ Solving for $\phi$: $$\phi = \frac{NBA}{k}\cdot I$$ Since $N$, $B$, $A$, and $k$ are all fixed constants of the instrument, the deflection $\phi$ is directly proportional to the current $I$. This linearity is what makes the galvanometer so useful — the scale is uniform and easy to calibrate.

Current sensitivity $S_I$ is defined as the deflection per unit current: $$S_I = \frac{\phi}{I} = \frac{NBA}{k}$$ (units: rad/A or divisions/μA). To increase sensitivity, designers use a stronger magnet (larger $B$), more turns $N$, larger coil area $A$, or weaker springs (smaller $k$). Voltage sensitivity $S_V$ is the deflection per unit voltage: $$S_V = \frac{\phi}{V} = \frac{NBA}{kR_G}$$ where $R_G$ is the resistance of the galvanometer coil. Interestingly, increasing $N$ increases $R_G$ proportionally, so $S_V$ may not improve simply by adding turns — a design trade-off engineers must consider carefully.

A galvanometer has a small full-scale deflection current (typically 1–100 μA) and cannot directly measure large currents or voltages. To make an ammeter, a low-resistance shunt $S$ is connected in parallel: $$S = \frac{I_g R_G}{I - I_g}$$ where $I_g$ is the full-scale deflection current and $I$ is the desired range. To make a voltmeter, a high-resistance multiplier $R_m$ is connected in series: $$R_m = \frac{V}{I_g} - R_G$$ These conversions illustrate the galvanometer's versatility as the core sensing element of classical electrical measurement.

$S_I = 5000\ \text{rad/A},\quad I_g \approx 1.047 \times 10^{-4}\ \text{A},\quad S \approx 2.62 \times 10^{-3}\ \Omega$

• Force on a Current-Carrying Conductor — The galvanometer's deflecting torque arises directly from the force $F = BIl$ on the current-carrying sides of the coil — understanding this force law is prerequisite to understanding galvanometer operation.
• Torque on a Current Loop in a Magnetic Field — The deflection of the galvanometer coil is caused by the torque $\tau = NBIA\sin\theta$ acting on the rectangular current loop; in a radial field $\sin\theta = 1$, making deflection linear.
• Conversion of Galvanometer to Ammeter and Voltmeter — Shunt and multiplier resistor calculations directly extend galvanometer theory to practical electrical measurements, forming the basis of classical analog metering circuits.
• Permanent Magnets and Magnetic Fields — The strong radial magnetic field in the galvanometer gap is produced by a carefully shaped permanent magnet; understanding magnetic field geometry explains why the deflection scale is linear.
• Restoring Forces and Torsional Springs — The phosphor-bronze springs provide a restoring torque proportional to deflection angle ($\tau_r = k\phi$), analogous to Hooke's Law — a concept from mechanics that governs the equilibrium position of the pointer.