Ampère’s Law & 3rd Maxwell Equation

In this lesson, we introduce Ampère’s Circuital Law and use it to derive the third Maxwell equation in magnetostatics. We begin by stating Ampère’s law in its integral form, which relates the line integral of the magnetic field intensity H around a closed path to the total steady current enclosed by that path. The physical meaning of the law is emphasized, highlighting its role as the magnetic counterpart to Gauss’s law in electrostatics. Using Stokes’ theorem, Ampère’s law is then converted from its integral form to its differential form, leading to the third Maxwell equation: ∇×H=J This result provides a local, pointwise relationship between magnetic fields and current density. A worked example demonstrates the application of Ampère’s law to find the magnetic field produced by an infinite straight current-carrying conductor, showing how symmetry simplifies the analysis and leads to a closed-form solution.