12.4 Ampère’s Law

Pick any closed path, called an Amperian loop. Go around it, adding up the part of $\vec{B}$ that points along the path times each small step: $\oint \vec{B} \cdot d\vec{\ell}$. Ampère's law says this equals $\mu_0$ times the enclosed current, the net current passing through any surface bounded by the loop: $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{enc}$.

Signs come from the right-hand rule: curl your fingers the way you go around the loop, and current along your thumb counts as positive. Currents outside the loop count for nothing in the total, even though they do change the field along it.

Ampère's law is always true, but it only helps you find B when symmetry makes B constant along parts of the loop and either parallel or perpendicular to the path. This is the magnetic partner of Gauss's law (8.6).

For a long straight wire, choose a circle of radius r centered on the wire. B has the same size everywhere on the circle and points along it, so $\oint \vec{B} \cdot d\vec{\ell} = B(2\pi r) = \mu_0 I$, giving $B = \dfrac{\mu_0 I}{2\pi r}$.

Inside a solid wire of radius R with current spread evenly, the loop encloses only part of the current: $I_{enc} = I\dfrac{r^2}{R^2}$. Then $B = \dfrac{\mu_0 I r}{2\pi R^2}$, which grows linearly from zero at the center to its maximum at the surface.

If the current density varies with r, find the enclosed current with the integral from 11.1, $I_{enc} = \displaystyle\int_0^r J(r')\,2\pi r'\,dr'$, then use $B(2\pi r) = \mu_0 I_{enc}$.

A solenoid is a long coil of wire. Inside a long solenoid the field is nearly uniform and points along the axis; outside it's nearly zero. A rectangular Amperian loop with one side inside gives $B\ell = \mu_0 (n\ell) I$, so $B = \mu_0 nI$, where n is the number of turns per unit length (N/ℓ). The field doesn't depend on the radius or on where you are inside.

For a wide flat conducting slab carrying a current density, the field outside is uniform, parallel to the slab and perpendicular to the current, pointing opposite ways on the two sides. Use a rectangular loop of length $\ell$ that straddles the slab, with its long sides parallel to the slab. Only those two sides add to $\oint \vec{B} \cdot d\vec{\ell}$, and the enclosed current is the current density times the cross-sectional area the loop surrounds. For a slab of thickness d carrying a uniform current density J, $2B\ell = \mu_0 Jd\ell$, so outside $B = \dfrac{\mu_0 Jd}{2}$, the same at any distance. Inside, at distance x from the slab's middle plane, the loop encloses only a thickness 2x, so $B = \mu_0 Jx$.

When several currents are present, find each one's field at the point and add the vectors. For two parallel wires, the fields add between the wires when the currents are opposite and partly cancel when the currents are the same.

Maxwell realized the law needed one more piece: a changing electric field also creates a magnetic field. That's why a magnetic field circles the gap between the plates of a charging capacitor, even though no charge crosses the gap. You need this idea only qualitatively. With it, Ampère's law joins Gauss's laws and Faraday's law (13.2) as one of Maxwell's equations.