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A current loop has magnetic moment $\vec{m} = NI\vec{A}$ (direction by right-hand rule). In a uniform field: torque $\vec{\tau} = \vec{m} \times \vec{B}$, magnitude $\tau = mB\sin\theta$. Potential energy $U = -\vec{m} \cdot \vec{B} = -mB\cos\theta$. Stable equilibrium at $\theta = 0$ ($\vec{m} \parallel \vec{B}$, $U_{\min} = -mB$); unstable at $\theta = \pi$ ($\vec{m}$ antiparallel, $U_{\max} = +mB$). Work to rotate from $\theta_1$ to $\theta_2$: $W = mB(\cos\theta_1 - \cos\theta_2)$.

The moving coil galvanometer exploits this torque. A coil of $N$ turns, area $A$, in radial field $B$ (created by concentric pole pieces) experiences $\tau = NIAB$ at all angles (since $\sin\theta = 1$ always in radial geometry). At equilibrium: $NIAB = k\phi$, so $\phi \propto I$ — giving a linear scale. Current sensitivity: $\phi/I = NAB/k$. Voltage sensitivity: $\phi/V = NAB/(kR)$.

To increase sensitivity: large $N$, $A$, $B$; small $k$. Converting to ammeter: add low shunt $S = GI_g/(I-I_g)$ in parallel. Converting to voltmeter: add high resistance $R_s = V/I_g - G$ in series. Ideal ammeter: zero resistance. Ideal voltmeter: infinite resistance.