Biot-Savart Law

The Biot-Savart Law relates magnetic fields to the currents which are their sources. In a similar manner, Coulomb's law relates electric fields to the point charges which are their sources. Finding the magnetic field resulting from a current distribution involves the vector product, and is inherently a calculus problem when the distance from the current to the field point is continuously changing.

We now use the Biot-Savart law to deal with problems in magnetostatics: this is the situation of steady currents leading to constant magnetic fields. There are two simple cases where the magnetic field integrations are easy to carry out, and fortunately they are in geometries that are of practical use. We use the formula for the magnetic field of an infinitely long wire whenever we want to estimate the field near a segment of wire, and we use the formula for the magnetic field at the center of a circular loop of wire wheneverwe want to estimate the magnetic field near the center of any loop of wire.

Infinitely Long Wire: The magnetic field at a point a distance r from an infinitely long wire carrying current I has magnitude and its direction is given by a right-hand rule: point the thumb of your right hand in the direction of the current, and your fingers indicate the direction of the circular magnetic field lines around the wire.

Circular Loop: The magnetic field at the center of a circular loop of current-carrying wire of radius R has magnitude and its direction is given by another right-hand rule: curl the fingers of your right hand in the direction of the current flow, and your thumb points in the direction of the magnetic field inside the loop.

Long Thick Wire: Imagine a very long wire of radius a carrying current I distributed symmetrically so that the current density, J, is only a function of distance r from the center of the wire. Ampere's law can be used to find the magnetic field at any radius r. Outside the wire, where

, we have just as if all the current were concentrated at the center of the wire. Inside the wire, where r < a, we have

where I(r) is the current flowing through the disk of radius r inside the wire; the current outside this disk contributes nothing to the magnetic field at r. Note that this is analogous to the result for symmetric electric fields.

Long Solenoid: Imagine a long solenoid of length L with N turns of wire wrapped evenly along its length. Ampere's law can be used to show that the magnetic field inside the solenoid is uniform throughout the volume of the solenoid (except near the ends where the magnetic field becomes weak) and is given by

where n = N/L.

Toroid: Imagine a toroid consisting of N evenly spaced turns of wire carrying current I. Ampere's law can be used to show that the magnetic field within the volume enclosed by the toroid is given by