Biot Savart Law Notes
The magnetic equivalent of Coulomb’s law is the Biot-Savart law for the magnetic field produced by a short segment of wire, , carrying current I:
where the direction of is in the direction of the current and where the vector points from the short segment of current to the observation point where we are to compute the magnetic field. Since current must flow in a circuit, integration is always required to find the total magnetic field at any point. The constant is chosen so that when the current is in amps and the distances are in meters, the magnetic field is correctly given in units of tesla. Its value in our SI units is exactly
A quick comparison of this value with the Biot-Savart law probably makes you wonder what role is supposed to play here. It plays the same role it did in Coulomb’s law: it was required in Coulomb’s law so that Gauss’s law wouldn’t have a , and it is required in the Biot-Savart law so that Ampere’s law won’t have one either.
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 whenever we 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, discussed in Chapter 24.
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 currentI. (Imagine winding wire onto a bagel, with the wire coming up through the hole, out around the outside, then up through the hole again, etc..) Ampere’s law can be used to show that the magnetic field within the volume enclosed by the toroid is given by
where R is the distance from the z-axis in cylindrical coordinates, with the z-axis pointing straight up through the hole in the center of the bagel.