Summary

This sub-topic developed WBCHSE Unit 3's first half, moving charges and magnetism, from

Oersted's original discovery through to a working ammeter and voltmeter. Oersted's experiment

(Section 4.2) first showed that a current-carrying wire produces a magnetic field circling around it

(right-hand thumb rule), unifying electricity and magnetism for the first time. The Biot-Savart law, $d\vec{B} = (\mu_0/4\pi)\,I\,d\vec{l}\times\hat{r}/r^2$ with $\mu_0/4\pi=10^{-7} \ \text{T}\,\text{m/A}$ (Section 4.3), gives the field due to a current element, and integrating it

gives the field of a long straight wire, $B=\mu_0I/2\pi r$ (Section 4.4), and of a circular loop,

$B_{\text{centre}}=\mu_0NI/2R$ and $B_{\text{axis}}(x)=\mu_0IR^2/[2(R^2+x^2)^{3/2}]$

(Section 4.5). Ampere's circuital law, $\oint\vec{B}\cdot d\vec{l}=\mu_0I_{\text{enc}}$

(Section 4.6), re-derives the straight-wire result in a few lines and, applied to a long

solenoid, gives its interior field $B=\mu_0nI$, uniform and independent of radius or axial

position (Section 4.7). Turning to the effect of a field on a moving charge, the Lorentz force

$\vec{F}=q[\vec{E}+(\vec{v}\times\vec{B})]$ (Section 4.8) combines the familiar electric force with

a magnetic force that can change direction but never speed; perpendicular to a uniform field, this

produces circular motion of radius $r=mv/qB$ and a speed-independent cyclotron frequency

$f=qB/2\pi m$, the operating principle of the cyclotron (Section 4.9). A current-carrying

conductor feels a force $F=BIL\sin\theta$ (Section 4.10), and two such conductors, placed parallel,

exert a mutual force per unit length $F/l=\mu_0I_1I_2/2\pi d$ -- same-direction currents attract,

opposite repel -- which historically DEFINED the ampere (Section 4.11). A current loop in a uniform

field feels a torque $\tau=NIAB\sin\theta=|\vec{m}\times\vec{B}|$ (Section 4.12), exploited …