Ampere's Circuital Law

Line integral of B around any closed Amperian loop equals μ₀ times net enclosed current. Valid for steady currents. Most useful for high-symmetry current distributions.

Statement

For any closed Amperian loop in a magnetostatic field:

$$\boxed{\oint \vec{B} \cdot d\vec{l} = \mu_0\,I_{\text{enc}}}$$

$I_{\text{enc}}$ is the net current threading the loop, counted algebraically.

Analogy with Gauss's Law

Sign Convention for $I_{\text{enc}}$

Curl the right hand in the direction of traversal of the loop. The thumb gives the positive normal. Current in the thumb direction is positive; current opposing it is negative.

Choosing the Amperian Loop

1. Identify the symmetry of the current distribution.
2. Choose a loop shape where $\vec{B} \parallel d\vec{l}$ and $|\vec{B}|$ is constant on each segment.
3. Then $\oint \vec{B} \cdot d\vec{l} = B \cdot L_{\text{eff}}$ — simple algebra.

Limitations

Valid only for steady currents. For time-varying fields, Maxwell added the displacement current term $\mu_0\varepsilon_0\,\partial\vec{E}/\partial t$ to the right side.