### 29.9: Ampere's Law

A fundamental property of a static magnetic field is that it is not conservative, unlike an electrostatic field. Instead, there is a relationship between the magnetic field and its source, electric current. Mathematically, this is expressed in terms of the line integral of the magnetic field, which is also known as Ampère’s law. It is valid only if the currents are steady and no magnetic materials or time-varying electric fields are present.

Ampère's law states that for any closed looped path, the line integral of the magnetic field along the path is proportional to the current enclosed in the loop. If the right-hand fingers curl along the direction of the integrating path, the current in the direction of the thumb is considered positive. The current opposite to the thumb direction is considered negative. If the integral of the magnetic field for a closed path is zero, it does not imply that the magnetic field is zero everywhere along the path; instead, the net current through the closed path is zero.

The electric field is easier to calculate for highly symmetric charge distributions using Gauss's law. Similarly, for highly symmetric current distributions, Ampère’s law can be used to evaluate the magnetic field. The line integral of the magnetic field along a closed path is known as the circulation of the magnetic field. Consider an infinitely long straight wire where the magnetic field surrounds the wire in a circular pattern. A small length element is parallel to the magnetic field along the Ampèrian loop and acts tangential to the path. Thus, the circulation of the magnetic field equals the constant magnetic field times the circumference of the circular path. Using Ampère’s law, the circulation of the magnetic field equals the permeability times the enclosed current.

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