# Amperes Law

**Amperes Law:**** **Biot-Savart's law for magnetic field due to a current element is difficult to visualize physically as such elements cannot be isolated from the circuit which they are part of. Andre Ampere formulated a law based on Oersted's as well as his own experimental studies. Ampere's law states that `` *the line integral of magnetic field around any closed path equals times the current which threads the surface bounded by such closed path. *. Mathematically,

In spite of its apparent simplicity, Ampere's law can be used to calculate magnetic field of a current distribution in cases where a lot of information exists on the behaviour of . The field must have enough symmetry in space so as to enable us to express the left hand side of (1) in a functional form. The simplest application of Ampere' s law consists of applying the law to the case of an infinitely long straight and thin wire.

**Ampere's Law in Differential Form:** We may express Ampere's law in a differential form by use of Stoke's theorem, according to which the line integral of a vector field is equal to the surface integral of the curl of the field,

The surface is *any * surface whose boundary is the closed path of integration of the line integral.

n terms of the current density , we have,

where is the total current through the surface . Thus, Ampere's law is equivalent to

which gives

You may recall that in the case of electric field, we had shown that the divergence of the field to be given by . In the case of magnetic field there are no free sources (monopoles). As a result the divergence of the magnetic field is zero

The integral form of above is obtained by application of the divergence theorem

Thus the flux of the magnetic field through a closed surface is zero.