29.2:
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.
29.2:
Biot-Savart Law
The Biot-Savart law gives the magnitude and direction of the magnetic field produced by a current. This empirical law was named in honor of two scientists, Jean-Baptiste Biot and Félix Savart, who investigated the interaction between a straight, current-carrying wire and a permanent magnet.
A current-carrying wire creates a magnetic field in its vicinity. Consider an infinitesimal current element dl in a wire. The direction of vector dl is along the direction of the current. The total magnetic field at point P due to all the charges in the current element is the vector sum of the field due to the individual charges. If A is the cross-sectional area, then Adl is the volume of the current element. Considering there are n number of charges per unit volume, nAqdl gives the total charge in the current element, while nqvdA is the current flowing through the wire. Substituting these, the magnitude of the magnetic field can be written in terms of the current as,
Defining a unit vector pointing from dl to P, the magnetic field due to the current element is given by the Biot-Savart law.
The magnetic field due to a finite length of current-carrying wire is found by integrating Equation 2 along the wire,
Suggested Reading
OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from 536-537 https://openstax.org/books/university-physics-volume-2/pages/12-1-the-biot-savart-law