Ampère's Circuital Law and Its Applications

The Concept of Circulation

Imagine placing a small paddle wheel in a flowing river. If the water flows in a straight line, the paddle wheel does not spin. However, if the water flows in a circular pattern, the paddle wheel rotates. In vector calculus, this rotation is quantified by the circulation of the velocity field around a closed loop. Similarly, magnetic fields often exhibit a "swirling" character around current-carrying conductors. The line integral of the magnetic field \(\vec{B}\) along a closed path \(C\) measures this total circulation. Ampère's law states that this circulation is not arbitrary; it is directly proportional to the net current enclosed by the path. If no current passes through the loop, the total circulation is zero, even if the field is non-zero at points along the path. This implies that the magnetic field is a solenoidal field generated by currents, distinct from the electrostatic field which is generated by charges.

Derivation from Biot–Savart

Although Ampère's law is often presented as a fundamental postulate, it is a direct consequence of the Biot–Savart law for steady currents. The Biot–Savart law gives the magnetic field \(\vec{B}\) at a point \(\vec{r}\) due to a current element \(I d\vec{l}'\) as:

To find the curl of \(\vec{B}\), we apply the vector identity \(\nabla \times (\vec{A} \times \vec{B}) = \vec{A}(\nabla \cdot \vec{B}) - \vec{B}(\nabla \cdot \vec{A}) + (\vec{B} \cdot \nabla)\vec{A} - (\vec{A} \cdot \nabla)\vec{B}\). After performing the integration over the current distribution and utilizing the fact that \(\nabla' \cdot \vec{J} = 0\) for steady currents, the result simplifies to the differential form of Ampère's law:

By applying Stokes' theorem, which states that \(\oint_C \vec{B} \cdot d\vec{l} = \int_S (\nabla \times \vec{B}) \cdot d\vec{a}\), we obtain the integral form:

This derivation confirms that Ampère's law is a theorem of magnetostatics, valid specifically for steady currents where the divergence of the current density is zero.

Application: Long Straight Wire

Consider an infinitely long straight wire carrying a steady current \(I\). Due to cylindrical symmetry, the magnetic field \(\vec{B}\) must be directed tangentially to circles centered on the wire, and its magnitude \(B\) depends only on the radial distance \(r\) from the axis. We choose a circular Amperian loop of radius \(r\) centered on the wire, lying in a plane perpendicular to it.

Along this loop, \(\vec{B}\) is parallel to \(d\vec{l}\) everywhere, so \(\vec{B} \cdot d\vec{l} = B \, dl\). The line integral becomes:

The current enclosed by this loop is simply \(I\). Applying Ampère's law:

Solving for \(B\), we obtain the familiar expression for the magnetic field of a long straight wire:

This result matches the one obtained by direct integration using the Biot–Savart law, but with significantly less computational effort.

Application: Solenoid and Toroid

For a long solenoid with \(n\) turns per unit length carrying current \(I\), we choose a rectangular Amperian loop that runs parallel to the axis inside the solenoid and returns outside. Inside the solenoid, the field is approximately uniform and parallel to the axis. Outside, the field is negligible for an ideal infinite solenoid. The contribution to the line integral comes only from the segment inside the solenoid of length \(L\). The number of turns enclosed is \(nL\), so the enclosed current is \(I_{\text{enc}} = nLI\). Thus:

For a toroid with \(N\) total turns and mean radius \(R\), we choose a circular Amperian loop of radius \(r\) within the toroidal core. The field is tangential and depends on \(r\). The enclosed current is \(I_{\text{enc}} = N I\) if the loop links all turns, or zero if it does not. For a loop inside the core:

These examples demonstrate the power of Ampère's law in determining fields where symmetry allows the field magnitude to be constant along the chosen path.