Ampere’s law related the integrated magnetic field around a closed loop to the electric current passing through the loop. It was discovered in 1826 by Andrew-Marie Ampere [1]. In was derived using hydrodynamics in 1861 by James Clerk Maxwell. Ampere’s law is now one of the Maxwell equations which form the basis of classical electromagnetism.

Using Ampere’s law, the magnetic field associated with a given current or current associated with a given magnetic field can be determined. This is provided that there is no time changing electric field presents. Ampere’s law can be written in two forms, integral and differential. These two forms are equivalent and can be related by the Kelvin-Stokes theorem. It can also be written in terms of B or H magnetic fields.

Simple electric current situations are classified when the electric current is “free current”. This is when the current passes through a wire or battery. In contrast, “bound current” is when bulk material can be magnetized and/or polarized.

All current is fundamentally the same. However, there are often practical reasons for wanting to treat bound current differently from free current. For instances, the bound current typically originates over atomic dimensions and one may wish to take advantage of a simpler theory intended for larger dimensions.

[1] Richard Fitzpatrick (2007). "Ampere's Circuital Law".