Torque on a Current Loop and the Moving Coil Galvanometer notes: ISC Class 12 Physics

1. Torque on a rectangular coil

Coil PQRS of N turns, current I, free to turn about an axis perpendicular to a uniform field B. Each side feels $\vec F = NI\,\vec l \times \vec B$ .
The two sides parallel to the axis (length l) feel forces NIlB, equal and opposite on different lines: a couple. The other two sides feel equal, opposite, collinear forces: no torque. Net force = 0 in a uniform field.
Arm of the couple $= b\sin\theta$ (θ between the normal and B): $\tau = NIlB \cdot b\sin\theta = NIAB\sin\theta$ , true for a flat coil of any shape.
Largest ( $NIAB$ ) when the plane of the coil is along B; zero when the plane is perpendicular to B.

Formulas

Torque on a coil
$\tau = NIAB\sin\theta$

Key terms

Couple: Two equal, opposite, parallel forces on different lines: no net force, but a torque equal to one force times the distance between the lines.

2. Magnetic moment and the vector torque

Magnetic moment $\vec m = NI\vec A$ (A m²), along the normal given by the right-hand rule: fingers curled along the current, thumb along $\vec m$ . Circular coil: $m = NI\pi R^2$ .
$\vec\tau = \vec m \times \vec B$ : the torque turns $\vec m$ towards $\vec B$ . θ = 0°: zero torque, stable; θ = 90°: largest torque; θ = 180°: zero torque, unstable.
If the plane of the coil makes angle α with B, then θ = 90° − α.

τ = NIAB sin θ

θ = 0°: stable equilibrium; θ = 180°: unstable equilibrium.

Formulas

Moment and torque
$\vec m = NI\vec A,\qquad \vec\tau = \vec m \times \vec B$

Key terms

Magnetic moment of a coil: NIA, a vector along the right-hand normal to the coil; unit A m².

3. Current loop as a magnetic dipole; the orbiting electron

A small current loop behaves as a magnetic dipole: field lines like those of a short bar magnet; the face $\vec m$ points out of is a north face. Far away on the axis $B \approx \dfrac{\mu_0}{4\pi}\dfrac{2m}{x^3}$ (compare $\dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{x^3}$ for an electric dipole). In a uniform field: no net force, torque $\vec m \times \vec B$ .
Electron in a circular orbit (Bohr model): $I = \dfrac{ev}{2\pi r}$ , $\mu = I\pi r^2 = \dfrac{evr}{2} = \dfrac{e}{2m_e}L$ ; $\vec\mu$ is opposite to $\vec L$ (negative charge). Gyromagnetic ratio $\dfrac{e}{2m_e} \approx 8.8 \times 10^{10}$ C kg⁻¹.
Bohr orbit $L = \dfrac{nh}{2\pi}$ : $\mu = n\mu_B$ , Bohr magneton $\mu_B = \dfrac{eh}{4\pi m_e} \approx 9.27 \times 10^{-24}$ A m².

Formulas

Orbital magnetic moment
$\mu = \dfrac{evr}{2} = \dfrac{e}{2m_e}L$
Bohr magneton
$\mu_B = \dfrac{eh}{4\pi m_e}$

Key terms

Bohr magneton: Orbital magnetic moment of the electron in the first Bohr orbit, eh/4πmₑ ≈ 9.27 × 10⁻²⁴ A m².
Gyromagnetic ratio: Ratio of orbital magnetic moment to orbital angular momentum, e/2mₑ for an electron.

4. Moving coil galvanometer

Construction: coil of many turns of fine insulated wire on a light frame; strong magnet with concave pole pieces; soft-iron cylinder inside the coil; phosphor-bronze suspension (or hairsprings) giving the restoring torque; mirror and lamp-and-scale, or a pointer.
Principle: a current-carrying coil in a magnetic field experiences a torque.
Radial field: the plane of the coil always contains B, so the deflecting torque is $NIAB$ at every deflection. The soft iron also makes B stronger.
Working: $NIAB = c\theta \Rightarrow I = \dfrac{c}{NAB}\theta = k\theta$ ( $c$ = restoring torque per unit twist, $k$ = galvanometer constant): a linear scale.
Current sensitivity $\dfrac{\theta}{I} = \dfrac{NAB}{c}$ ; voltage sensitivity $\dfrac{\theta}{V} = \dfrac{NAB}{cG}$ . More turns raise both N and G.

radial field: I = kφ
uniform field: I ∝ φ/cos φ

Only the radial field gives a linear scale.

Formulas

Galvanometer
$I = \dfrac{c}{NAB}\,\theta = k\theta$
Sensitivities
$\dfrac{\theta}{I} = \dfrac{NAB}{c},\qquad \dfrac{\theta}{V} = \dfrac{NAB}{cG}$

Key terms

Radial field: Field directed along radii of the soft-iron core, so the plane of the coil always contains B.
Current sensitivity: Deflection per unit current, NAB/c.
Voltage sensitivity: Deflection per unit p.d. across the coil, NAB/cG.

5. Ammeter and voltmeter from a galvanometer

Ammeter of range I: shunt in parallel, $I_gG = (I - I_g)S \Rightarrow S = \dfrac{I_gG}{I - I_g}$ ; for $I = nI_g$ , $S = \dfrac{G}{n - 1}$ . Resistance $\dfrac{GS}{G + S}$ (very small); connected in series.
Voltmeter of range V: resistance in series, $V = I_g(G + R) \Rightarrow R = \dfrac{V}{I_g} - G$ . Resistance $G + R$ (very large); connected in parallel. Taps give several ranges.
Ideal ammeter: zero resistance; ideal voltmeter: infinite resistance. A real voltmeter across a resistor comparable with its own resistance reads low.

Worked example 6 values.

Worked example 7 values.

reading

A good voltmeter has a resistance much larger than the resistance it is connected across.

Formulas

Shunt
$S = \dfrac{I_gG}{I - I_g}$
Series resistance
$R = \dfrac{V}{I_g} - G$

Key terms

Shunt: Small resistance connected in parallel with a galvanometer to carry most of the current.

Exam tips

Torque: θ is measured from the normal to the coil; the plane at α means θ = 90° − α.
Direction of m: right-hand rule only.
Galvanometer: radial field gives torque NIAB at every angle, so I = kθ.
Shunt: equal p.d. across coil and shunt. Voltmeter: total resistance V/Ig.

1. Torque on a rectangular coil

Formulas

Key terms

2. Magnetic moment and the vector torque

Formulas

Key terms

3. Current loop as a magnetic dipole; the orbiting electron

Formulas

Key terms

4. Moving coil galvanometer

Formulas

Key terms

5. Ammeter and voltmeter from a galvanometer

Formulas

Key terms

Exam tips

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