4.1.10 Recap

Lorentz force, Eq. (4.1.2): $F=q(E+v\times B)$ $[\mathrm{N}]$. The magnetic part is perpendicular to $v$ and to $B$, proportional to the speed, and does no work. On a wire: $F=Il\times B$, with $l$ along the current. $B$ is an axial vector, a sense of rotation rather than an arrow.

Current density: $J=\rho v$ $[A/{\mathrm{m}}^2]$, $I_S={\int }_{S}J\cdot dS$. Volume, surface ($J_S$ $[A/m]$) and line ($I$ $[\mathrm{A}]$) currents, $Jd{v}^{\prime }\to J_{S}d{S}^{\prime }\to Id{l}^{\prime }$. Moving charge is current.

Biot–Savart's law, Eq. (4.1.10): $B=\frac{{\mu }_0}{4\pi }\int J\times Rd{v}^{\prime }/R^3$, $R$ from source to field point, ${\mu }_0=4\pi \times 10^{-7}$ H/m. $H=B/{\mu }_0$ $[A/m]$ is fixed by the currents alone. Each element gives a field circling it, $\propto 1/R^2$.

Vector potential, Eq. (4.1.15): $A=\frac{{\mu }_0}{4\pi }\int Jd{v}^{\prime }/R$ $[Wb/m]$, parallel to the current, and $B=\nabla \times A$.

Fundamental equations, Eqs. (4.1.18) and (4.1.16):

Currents make $H$ curl, nothing makes $B$ diverge: field lines never end.

Magnetic flux: $\Phi ={\int }_{S}B\cdot dS={\oint }_{C}A\cdot dl$ $[\mathrm{Wb}]$, the same through every surface with the same rim.

Standard fields: long wire $H=I/2\pi \rho$ along $a_{\varphi }$ (right hand), loop axis $B_z={\mu }_{0}I{b}^2/2(b^2+h^2{)}^{3/2}$, centre ${\mu }_{0}I/2b$, far field ${\mu }_{0}m/2\pi h^3$ with $m=I\pi b^2$.

Ampère's law at work: symmetry $\Rightarrow$ surviving component $\Rightarrow$ Amperian path (coaxial circle, straddling rectangle, toroidal circle) $\Rightarrow$ $I_{\mathrm{enc}}$ $\Rightarrow$ check continuity, limits, units. Without an Amperian path: rings and the on-axis loop field.

Applications: the Hall voltage $V_H=IB/nqt$, the motor torque $NIAB\sin \alpha$, the $1/\rho$ field under a line and the clamp meter $I=\oint H\cdot dl$, and the pancake coil $H_z(0)=(NI/2b)\ln 2$.