Magnetism — Fields, Ampere & Charges in Fields (Medium)

Quick Recap — Magnetic Fields from Currents

• Sources ($\mu_0=4\pi\times10^{-7}$): long straight wire $B=\dfrac{\mu_0 I}{2\pi r}$; loop centre $B=\dfrac{\mu_0 N I}{2R}$; solenoid $B=\mu_0 n I$; arc of angle $\theta$ at its centre $B=\dfrac{\mu_0 I\theta}{4\pi R}$ (semicircle $\dfrac{\mu_0 I}{4R}$).
• Ampere's law: $\oint \vec B\cdot d\vec l=\mu_0 I_{enc}$.
• Charge in a field: $\vec F=q\vec v\times\vec B$ — always perpendicular to $\vec v$, so it does no work (speed and KE stay constant). Circular motion: $r=\dfrac{mv}{qB}$, period $T=\dfrac{2\pi m}{qB}$ (independent of $v$), frequency $f=\dfrac{qB}{2\pi m}$.
• Velocity selector: undeflected when $qE=qvB\Rightarrow v=\dfrac{E}{B}$.

Worked mini-example. A proton ($m=1.67\times10^{-27}$ kg) at $v=10^6$ m/s in $B=0.5$ T moves on a circle of radius $r=\dfrac{mv}{qB}=\dfrac{(1.67\times10^{-27})(10^6)}{(1.6\times10^{-19})(0.5)}\approx2.1$ cm.