Moving Charges and Magnetic Field — Notes

• Oersted (1820): a compass needle near a current-carrying wire deflects, and the deflection reverses with the current: a current produces a magnetic field. Field lines of a straight wire are circles round it (right-hand grip rule).
• A charge q moving with velocity v in a field B feels $\vec F = q(\vec v \times \vec B)$, of size $qvB\sin\theta$, perpendicular to both v and B. This defines B: $B = F/(qv\sin\theta)$.
• Unit: tesla, 1 T = 1 N A⁻¹ m⁻¹ = 1 N s C⁻¹ m⁻¹; 1 gauss = 10⁻⁴ T. No force on a charge at rest or moving along B.

$d\vec B = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec l \times \vec r}{r^3}$, size $dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}$, with $\mu_0 = 4\pi \times 10^{-7}$ T m A⁻¹.

• $d\vec B$ is perpendicular to the plane of $d\vec l$ and $\vec r$; zero along the line of the element.
• Compared with Coulomb’s law: both inverse-square and both obey superposition; but the source $I\,d\vec l$ is a vector, the field depends on $\sin\theta$ and is perpendicular to $\vec r$ (the electric field is along $\vec r$).

• Centre of a coil of N turns, radius R: every element is at 90° to the line to the centre and at distance R, so $B = \dfrac{\mu_0NI}{2R}$ along the axis (right-hand grip rule).
• Axis, distance x from the centre: components across the axis cancel in pairs; $B = \dfrac{\mu_0NIR^2}{2(R^2 + x^2)^{3/2}}$. Far away, $B \approx \mu_0NIR^2/2x^3$.
• Example: 40 turns, R = 8.0 cm, 0.60 A gives 1.88 × 10⁻⁴ T at the centre and 0.512 of that at x = 6.0 cm.

• $\oint \vec B \cdot d\vec l = \mu_0 I_{\text{enclosed}}$ round any closed loop; currents outside do not count, opposite currents count with opposite signs. Useful when symmetry makes B constant along the loop.
• Long straight wire: $B = \mu_0I/2\pi r$, circles round the wire.
• Long solenoid: $B = \mu_0nI$ inside (n = turns per metre), uniform and along the axis; weak outside; independent of the radius.
• Finite straight wire (formula only): $B = \dfrac{\mu_0I}{4\pi a}(\sin\alpha + \sin\beta)$, angles from the perpendicular; long wire $\alpha = \beta = 90^\circ$.

• $\vec F = q(\vec E + \vec v \times \vec B)$. The magnetic part is perpendicular to v, so it does no work: speed and kinetic energy stay constant.
• $\vec v \perp \vec B$: circle, $r = mv/qB$, $T = 2\pi m/qB$ (independent of v), $f = qB/2\pi m$.
• At angle θ: helix, $r = mv\sin\theta/qB$, pitch $p = 2\pi mv\cos\theta/qB$. Along B: straight line.
• Crossed E and B: undeflected when $v = E/B$ (velocity selector).

• Conductor of length l: $\vec F = I(\vec l \times \vec B)$, $F = BIl\sin\theta$ (from $nAl$ electrons each feeling $ev_dB\sin\theta$, with $I = nAev_d$). Zero when the wire is along B.
• Parallel wires: $f = \dfrac{\mu_0I_1I_2}{2\pi r}$ per unit length; like currents attract, unlike repel; equal and opposite forces.
• Ampere: the steady current in each of two long, thin, parallel wires 1 m apart in vacuum that gives a force of 2 × 10⁻⁷ N per metre. Coulomb: the charge carried in 1 s by 1 A (1 C = 1 A s).