Magnetic field of a conducting plane

Besides wires, loops and solenoids, there is another important geometry: a conducting sheet (infinite plane) carrying a uniformly distributed current. It is the magnetic analogue of the charged plane in electrostatics.

The distribution is described by the surface current density $K$, measured in A/m (amperes per metre of width): cutting the sheet with a line of length $\ell$ perpendicular to the current, a current $i = K\ell$ flows through it.

Flat conducting sheet with surface current density $\vec{K}$: the field is parallel to the sheet and perpendicular to $\vec{K}$, with opposite directions above and below.

Calculation with Ampère’s theorem. We choose a rectangular Ampèrian path of width $w$ and height $2h$, centred on the sheet, with the horizontal sides parallel to $\vec{K}$. By symmetry the field above and below is horizontal and of equal magnitude $B$; the vertical sides of the rectangle contribute nothing ($\vec{B}\perp\dd\vec{\ell}$). What remains are the two horizontal sides, which both contribute $B\,w$ in the same direction:

$\oint \vec{B}\cdot\dd\vec{\ell} = B\,w + B\,w = 2B\,w$

The enclosed current is $i_\text{conc} = K\,w$, so $2B\,w = \mu_0 K\,w$:

$\ev{B = \frac{\mu_0\,K}{2}}$

Law — Field of a flat sheet with current

An infinite conducting sheet with surface current density $K$ (in A/m) produces a uniform magnetic field on each side: $B = \frac{\mu_0\,K}{2}$ The field is parallel to the sheet, perpendicular to $\vec{K}$, and changes direction going from one side to the other. Notably: the field does not depend on the distance from the sheet.

Electrostatic analogy

Just as the charged plane gives $E = \sigma/(2\varepsilon_0)$ (uniform, independent of distance), the plane with current $K$ gives $B = \mu_0 K/2$ on each side.

Example — Two parallel sheets with opposite currents

Two parallel sheets carry current density $K$ in opposite directions, like the plates of a capacitor. Between them the fields add up, outside they cancel. In the internal region: $B_\text{interno} = 2\cdot\frac{\mu_0 K}{2} = \mu_0 K$ This is the configuration analogous to the flat solenoid, used in “ribbon” coaxial cables.

Collegamenti

Argomenti: Magnetismo Concetti: Teorema di Ampère · Campo magnetico

Esercizi collegati: Dimensional check of the solenoid field · Field inside a solenoid · Problem — B field in a charging capacitor