Moving Charges and Magnetism — Notes

Lorentz force: $\vec F = q(\vec E + \vec v \times \vec B)$. Magnetic part $qvB\sin\theta$, perpendicular to $\vec v$ and $\vec B$; zero for $v = 0$ or $\vec v \parallel \vec B$; does no work (speed constant). Unit of $B$: tesla (T).

Uniform $\vec B$, $\vec v \perp \vec B$: circle with $r = \dfrac{mv}{qB}$, period $T = \dfrac{2\pi m}{qB}$, frequency $f = \dfrac{qB}{2\pi m}$ — independent of speed. With a component of $\vec v$ along $\vec B$: a helix (pitch $= v_\parallel T$).

Crossed fields (velocity selector): $\vec E \perp \vec B \perp \vec v$ with opposing forces; undeflected when $qE = qvB$, i.e. $v = E/B$, for any charge or mass.

Biot–Savart law: $d\vec B = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec l \times \hat r}{r^2}$, $dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}$, $\mu_0 = 4\pi \times 10^{-7}$ T m A$^{-1}$.

Axis of a circular loop (radius $R$, distance $x$): each element is at distance $\sqrt{R^2+x^2}$ and perpendicular to the line to P, so $dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl}{R^2+x^2}$. Perpendicular components cancel in pairs; axial components $dB\cos\alpha$ with $\cos\alpha = R/\sqrt{R^2+x^2}$ add over $\oint dl = 2\pi R$: $B = \dfrac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$ At the centre $B = \mu_0 I/2R$ ($N$ turns: multiply by $N$). Far away ($x \gg R$): $B \approx \dfrac{\mu_0}{4\pi}\dfrac{2m}{x^3}$ with $m = I\pi R^2$. Direction: right-hand rule (fingers with the current, thumb along $\vec B$).

• Long straight wire: circle of radius $r$ round the wire; $B$ tangential and constant: $B = \dfrac{\mu_0 I}{2\pi r}$. Field lines are concentric circles (right-hand thumb rule).
• Long solenoid ($n$ turns per metre): uniform field inside along the axis, $B = \mu_0 nI$; nearly zero outside; behaves like a bar magnet (treated qualitatively in the syllabus).

Like Gauss’s law, Ampere’s law always holds but gives $B$ easily only when symmetry makes $B$ constant along the chosen loop.

Force on a straight conductor: $\vec F = I\vec l \times \vec B$, $F = IlB\sin\theta$.

Two long parallel wires a distance $d$ apart: $\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}$. Currents in the same direction attract; opposite currents repel. The forces on the two wires are equal and opposite.

Ampere (classic definition): the current which, in each of two long parallel wires 1 m apart in vacuum, produces a force of $2 \times 10^{-7}$ N per metre of length.

A coil of $N$ turns, area $A$, current $I$ in a uniform field: net force zero; torque $\tau = NIAB\sin\theta,\qquad \vec\tau = \vec m \times \vec B,\qquad \vec m = NI\vec A$ ($\theta$ between the normal to the coil and $\vec B$). Maximum when the plane of the coil is parallel to $\vec B$; zero when it is perpendicular. Stable equilibrium with $\vec m$ along $\vec B$.

A current loop is a magnetic dipole (moment in A m²), the analogue of an electric dipole: $\vec\tau = \vec p\times\vec E$ ↔ $\vec\tau = \vec m\times\vec B$; far field $\propto m/r^3$.

Coil on a soft-iron core in a radial field (concave pole pieces), so the torque is $NIAB$ at all deflections; spring restoring torque $k\phi$. Equilibrium: $\phi = \dfrac{NAB}{k}I$ — linear scale.

• Current sensitivity $\phi/I = NAB/k$; voltage sensitivity $\phi/V = NAB/kG$.
• Raising $N$ raises current sensitivity but also $G$, so voltage sensitivity may not rise.

Conversions ($G$, full-scale $I_g$):

• Ammeter of range $I$: shunt $S = \dfrac{I_gG}{I - I_g}$ in parallel (very low total resistance; connected in series).
• Voltmeter of range $V$: $R = \dfrac{V}{I_g} - G$ in series (very high total resistance; connected in parallel).