Biot-Savart Law

Moving charges constitute a current. The magnitude and direction of the magnetic field produced by a steady current are given by the Biot-Savart law.  

In a current-carrying wire, consider an infinitesimal charge element. The total charge in the element is given by the number of charges and the volume of the element.

The magnetic field at a point P due to all the charges in the element is the vector sum of the fields created by separate charges. The magnitude of the total magnetic field due to the charge element can be obtained by defining the angle between the drift velocity and the line joining point P.

Rewriting the equation in terms of the current gives the magnitude of the magnetic field created by the charge element.

Defining a unit vector along the line joining P and a vector along the length element, the expression for magnetic field is derived. 

Integrating the equation, the magnetic field due to a current-carrying wire of a finite length can be obtained using the Biot-Savart law.