Magnetism — Forces, Torque, Materials & Earth (Medium)

Quick Recap — Forces, Torque, Magnetic Materials & Earth

• Force on a conductor: $\vec F=I\vec L\times\vec B$, magnitude $BIL\sin\theta$. Parallel wires: $\dfrac{F}{L}=\dfrac{\mu_0 I_1 I_2}{2\pi d}$ — attract if the currents are parallel, repel if antiparallel.
• Current loop: moment $m=NIA$; torque $\tau=mB\sin\theta=NIAB\sin\theta$ (maximum when the plane is parallel to $B$); energy $U=-mB\cos\theta$.
• Bar magnet: axial field $\dfrac{\mu_0}{4\pi}\dfrac{2M}{r^3}$, equatorial $\dfrac{\mu_0}{4\pi}\dfrac{M}{r^3}$ (ratio $2:1$).
• Materials: diamagnetic ($\chi$ small, $<0$), paramagnetic ($\chi$ small, $>0$, Curie $\chi\propto\tfrac1T$), ferromagnetic ($\chi\gg0$, hysteresis).
• Earth: dip $\tan\delta=\dfrac{B_V}{B_H}$; $B_H=B\cos\delta$; galvanometer conversions use a shunt (ammeter) or a series resistor (voltmeter).

Worked mini-example. Two long wires 10 cm apart each carry 10 A the same way: $\dfrac{F}{L}=\dfrac{\mu_0 I_1 I_2}{2\pi d}=\dfrac{(2\times10^{-7})(10)(10)}{0.1}=2\times10^{-4}$ N/m, attractive.