Ampère's law

Circulation of magnetic field

Ampère's law is a powerful tool for finding magnetic fields in highly symmetric current distributions. It relates the line integral of magnetic field around a closed path to the current passing through that path.

The magnetostatic form is

oint ec{B}cdot dec{ell}=mu_0 I_{enc}

The integral is around a closed loop called an Amperian loop. $I_{enc}$ is the net current passing through any surface bounded by that loop.

Meaning of the line integral

The quantity

oint ec{B}cdot dec{ell}

measures magnetic circulation around a path. It is not flux. Magnetic flux uses ec{B}cdot dec{A} through a surface. Ampère's law uses ec{B}cdot dec{ell} along a closed curve.

Magnetic fields tend to circulate around currents, and Ampère's law captures that structure.

Long straight wire

For a long straight wire carrying current $I$, choose a circular Amperian loop of radius $r$ centered on the wire. By symmetry, $B$ has constant magnitude on the loop and is tangent to it. Therefore

oint ec{B}cdot dec{ell}=B(2pi r)

Ampère's law gives

$B(2pi r)=mu_0 I$

so

B=rac{mu_0I}{2pi r}

This matches the Biot-Savart result.

Solenoid

A solenoid is a long coil of wire. Inside a long ideal solenoid, the magnetic field is approximately uniform. Outside, the field is relatively weak.

If the solenoid has $n$ turns per unit length and current $I$, Ampère's law gives

$B=mu_0 nI$

inside the ideal solenoid.

This result assumes the solenoid is long compared with its radius, so edge effects are negligible.

Toroid

A toroid is a coil wrapped into a donut shape. Inside the toroidal core, an Amperian circle of radius $r$ encloses $N$ turns carrying current $I$. Ampère's law gives

$B(2pi r)=mu_0NI$

so

B=rac{mu_0NI}{2pi r}

The magnetic field is largely confined inside the toroid.

When Ampère's law is useful

Ampère's law is always true for steady currents, but it is useful for finding $B$ only when symmetry makes the line integral simple. Good examples include long straight wires, infinite current sheets, long solenoids, and toroids.

If the field is not constant or not parallel to the path in a useful way, Biot-Savart or numerical methods may be better.

Sign and right-hand rule

The sign of $I_{enc}$ depends on the chosen direction around the loop. Curl the fingers of your right hand in the integration direction; your thumb gives the positive current direction through the loop.

This convention keeps the line integral and current sign consistent.

Beyond magnetostatics

Ampère's law in this form is incomplete for time-varying electric fields. Maxwell corrected it by adding the displacement current term. The full Maxwell-Ampère law is essential for electromagnetic waves.

For steady currents, however, the simpler form works well.

The big idea

Ampère's law relates magnetic field circulation to enclosed current. It is the magnetic analog of a symmetry-based field law, especially useful for wires, solenoids, and toroids. Its deeper significance becomes even greater when Maxwell's displacement current completes the law for time-dependent fields.