Ampère's Law

Ampère's law is simply a rule that says: walk around any closed loop, keep track of how much the magnetic field points along your path, and the total depends only on the current passing through the loop. It's named after André-Marie Ampère, the French physicist who worked out in the 1820s how currents push on each other. It's what lets you find the field of a solenoid or a thick wire in one line, and it's how a clamp meter reads a current without touching the wire. The formula $\oint_C \mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{\text{enc}}$ is just four pieces: $\oint_C$ (go all the way around the loop $C$), $\mathbf{B}\cdot d\mathbf{l}$ (how much field points along each little step), $\mu_0$ (the same constant as before) and $I_{\text{enc}}$ (the current poking through the loop). The circle on the integral sign looks scary, but it just means "all the way around". Bear with it.

A meter that never touches the wire. An electrician closes the jaws of a clamp meter around one wire and reads the current without cutting anything. The jaws form a closed loop, and the meter measures how much magnetic field circulates around it. That circulation depends only on the current passing through the loop. That's Ampère's law.

Checking it on a straight wire. The field of a long wire is $\frac{\mu_0 I}{2\pi r}$, circling it (The Biot–Savart Law). Walk any loop around the wire. Only the part of each step along the circle counts, and a step that sweeps an angle $d\phi$ around the wire at distance $r$ moves $r\,d\phi$ that way, so

The distance cancels! Think about why. Farther out the field is weaker, like $1/r$, but each step sweeps a smaller angle for the same length, so the two effects exactly balance. Around a loop that circles the wire once, $\phi$ grows by $2\pi$ and the circulation is $\mu_0 I$, whatever the loop's shape. Around a loop that misses the wire, $\phi$ swings up and comes back, and the circulation is $0$. Superposition extends this to any collection of currents: only the ones threading the loop count.

Signs. Curl your right hand's fingers along the direction you walk the loop; currents along your thumb count positive, and currents the other way negative.

Using it. Ampère's law always holds, but it only finds $\mathbf{B}$ when symmetry tells you the field's direction and that its size is constant along a loop you can choose, just as Gauss's law does for $\mathbf{E}$. The recipe:

1. Use symmetry to decide which way $\mathbf{B}$ points and what it can depend on.
2. Pick a loop along which $\mathbf{B}$ is either parallel to the path with constant size or perpendicular to it.
3. Write $\oint\mathbf{B}\cdot d\mathbf{l}$ in terms of $B$, count $I_{\text{enc}}$, and solve.

Inside a thick wire. For a wire of radius $R$ carrying $I$ spread evenly, a circle of radius $r < R$ encloses the fraction $r^2/R^2$ of the current: $B\cdot2\pi r = \mu_0 I\frac{r^2}{R^2}$, so $B = \frac{\mu_0 Ir}{2\pi R^2}$. The field grows linearly from zero at the center, peaks at the surface, then falls off as $1/r$.

Solenoids. A long, tightly wound coil with $n$ turns per meter has a uniform axial field inside and essentially none outside. Take a rectangular loop with one long side, of length $\ell$, inside along the axis and the opposite side outside. Only the inside side contributes, $B\ell$; the short sides are perpendicular to $\mathbf{B}$; and the loop encloses $n\ell$ turns:

The radius doesn't appear, and neither does where the inside side sits: the field is the same everywhere inside. That's why solenoids are the workhorse for making uniform fields.

Toroids. Bend the solenoid into a donut with $N$ turns. Circles inside the core enclose $NI$, so $B\cdot2\pi r = \mu_0 NI$ and $B = \frac{\mu_0 NI}{2\pi r}$. Circles outside enclose as much current going up as coming down, so $B = 0$ there. The field stays trapped in the core, which is why transformer and inductor cores are often rings.

The local law, via Stokes. For any surface $S$ bounded by $C$, Stokes' theorem and the definition of current give

These agree for every surface, however small, so the integrands must match: $\nabla\times\mathbf{B} = \mu_0\mathbf{J}$. The field swirls exactly where current flows (Curl and Divergence). Outside a wire $\nabla\times\mathbf{B} = \mathbf{0}$, yet the circulation around the wire isn't zero: the region around a wire has a hole in it, and a curl-free field on a region with a hole need not be conservative. Together with $\nabla\cdot\mathbf{B} = 0$ (there are no magnetic charges, so field lines never end), this is the magnetic half of Maxwell's Equations.

One gap. As written, the law holds for steady currents. Around a wire feeding a charging capacitor it gives different answers for different surfaces. Fixing that is the displacement current, which we'll get to in the last unit of the course.

A common mistake. Since Ampère's law is always true, it's tempting to think it always hands you $B$. It doesn't. You can only pull $B$ out of the integral when symmetry promises it's constant along your loop, which fails for a short wire or a short solenoid. In those cases you're back to Biot–Savart. Two smaller slips: using the total current instead of the enclosed current, and using $N$ where the solenoid formula needs $n = N/\ell$.

Clamp meters read current by measuring the circulation around their jaws, and ground-fault breakers trip when the circulation around a live and neutral pair isn't zero. Solenoids built on $B = \mu_0 nI$ drive door locks, fuel injectors and MRI magnets, which use superconducting coils with thousands of amp-turns per meter. Toroidal cores keep transformer fields from leaking, coaxial cables keep their fields between the conductors, and fusion reactors confine plasma in a toroid's ring-shaped field.