Loops, Galvanometers and Magnetism of Matter: Quick Revision

A brief note on Loops, Galvanometers and Magnetism of Matter. Read it before a test. Try each question at the end before you open its answer.

A current loop in a field

• Magnetic moment: $m = NIA$, along the normal (right-hand rule). Unit: A m².
• Torque: $\tau = NIAB\sin\theta = mB\sin\theta$, with $\theta$ between the normal and $\vec{B}$. Net force in a uniform field: zero.
• Energy: $U = -mB\cos\theta$. Stable at $0^\circ$, unstable at $180^\circ$.
• Work to turn the loop: $W = mB(\cos\theta_1 - \cos\theta_2)$.
• An orbiting electron: $m = \dfrac{evr}{2}$ and $\dfrac{m}{L} = \dfrac{e}{2m_e}$.

Galvanometer, ammeter and voltmeter

• Radial field: torque $NIAB$ at every angle. Balance with the spring: $\phi = \dfrac{NAB}{k}I$, so the scale is even.
• Sensitivity: current $\dfrac{NAB}{k}$, voltage $\dfrac{NAB}{kG}$.
• Ammeter: shunt in parallel, $S = \dfrac{I_gG}{I - I_g}$. Small resistance, joined in series.
• Voltmeter: resistance in series, $R = \dfrac{V}{I_g} - G$. Large resistance, joined in parallel.
• Ideal meters: an ammeter has zero resistance and a voltmeter has infinite resistance.

Magnets and the Earth

• Field lines: closed loops. $\oint \vec{B}\cdot d\vec{A} = 0$, so there are no monopoles.
• Short magnet: axis $\dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}$, equatorial line $\dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}$.
• Swings of a magnet: $T = 2\pi\sqrt{\dfrac{J}{mB}}$.
• The Earth: $B_H = B_E\cos\delta$, $B_V = B_E\sin\delta$, $\tan\delta = \dfrac{B_V}{B_H}$. Dip $0^\circ$ at the magnetic equator, $90^\circ$ at the poles.

Magnetic materials

• Definitions: $B = \mu_0(H + M)$, $M = \chi H$, $\mu_r = 1 + \chi$.
• Diamagnetic: $\chi$ small, negative. Pushed away weakly.
• Paramagnetic: $\chi$ small, positive, $\chi = \dfrac{C}{T}$. Drawn in weakly.
• Ferromagnetic: $\chi$ large, positive. Domains. Paramagnetic above the Curie temperature.
• Hysteresis: soft iron has a narrow loop (electromagnets, transformers). Steel has a wide loop (permanent magnets).

A worked example

A coil has $60$ turns, each of area $12\ \text{cm}^2$, and carries $1.4$ A in a radial field of $0.18$ T. The spring has $k = 2.0 \times 10^{-6}\ \text{N m/rad}$. Find the deflection for a current of $80\ \mu$A.

1. The factor. $NAB = 60 \times 12 \times 10^{-4}\ \text{m}^2 \times 0.18\ \text{T} = 1.30 \times 10^{-2}\ \text{N m/A}$.
2. The deflection. $\phi = \dfrac{NAB}{k}I = \dfrac{1.30 \times 10^{-2}}{2.0 \times 10^{-6}\ \text{N m/rad}} \times 80 \times 10^{-6}\ \text{A} = 0.52\ \text{rad}$, about $30^\circ$.

Practise with new numbers

Check yourself

1. A coil has $m = 0.40\ \text{A m}^2$ in a field of $0.25$ T. What is the greatest torque, and the work to turn it from stable to unstable equilibrium?

2. A galvanometer has $G = 50\ \Omega$ and $I_g = 1.0$ mA. What resistance in series makes a voltmeter for $0$ to $10$ V?

3. The field of a short magnet on its axis is $12\ \mu$T at a distance $r$. What is it on the equatorial line at the same distance?

4. Name the kind of material from its susceptibility: (a) $-1 \times 10^{-5}$, (b) $+2 \times 10^{-4}$, (c) $+800$.