PhysicsLAB: A Guide to Biot-Savart Law

Probably one of the hardest, and most confusing, of the four electromagnetic equations is the Biot-Savart Law (pronounced bee-yo-suh-var).

This law is easily seen as the magnetic equivalent of Coulomb's Law. What it basically states is that the magnetic field decreases with the square of the distance from a "point of current" or current segment. Where it differs is the fact that a point of current is much harder to achieve than a point charge.

As mentioned earlier, the Biot-Savart law deals with a current element. A current element is like a magnetic element in that it is the current multiplied by distance. However a current element, by its very definition, cannot exist in a single point. Therefore, we must take the derivative of the current element and integrate a path of point-current elements. Stay with me, this becomes less confusing as it goes on.

Initially, let's try to derive the Biot-Savart Law from its similarity to Coulomb's Law and other facts that we already know. First, we'll start with an expression for an electric field around a point charge based on Coulomb's Law:

If we exchange

q with I dl (I is always constant in a wire) and dl makes it a point-current element or current segment
E with dB (infinitesimals must be conserved),

then we get the very basics of the Biot-Savart Law.

Our next step will be to decide what expression will replace "k". Since k in Coulomb's Law is

, and

is always on the opposite side of the fraction with

on these laws, the "k" for Biot-Savart law should be

So we now have

One final consideration that we must consider is that the current element has something that a point charge doesn’t have -- a direction. Since a magnetic field is strongest when it is at right angles to the current, we have to include the cross product of the direction of the radius,

where

is the angle between r and I.

That wasn’t so hard, was it? You might want to take a breather before continuing. Rested? Then let's use the Biot-Savart Law to find the magnetic field around a current carrying wire and at the center of a current loop.

Magnetic Field Around a Current Carrying Wire

First we are going to find the magnetic field at a distance R from a long, straight wire carrying a current of I. To do this, we must determine the proper use of Biot-Savart.

Pulling out all of the terms that aren’t related to distance will give us

This wire is long, so we are going to pretend that it is infinite in length.

¹ Using symmetry principles, we are going to cut our wire in half and change our limits. Later, these symmetry properties will allow us to double our final B-field's value.

where

is the distance from the point-current element to the closest point of the wire to the point, and
R is the distance from the point to the wire, and
r is the distance from the point-current element to the point.

and

Now we use the trigonometric identity

to replace r².

Now we need to replace our differential dl. For this we use the trigonometric identity:

Now we substitute and integrate; but, because our differential has changed, so our limits must change. When

, q becomes infinitely small and approaches zero; when

, q approaches

You should now recognize this result from our previous lesson on Ampere's Law.

Magnetic Field at the Center of a Current-Carrying Loop

Let’s try something else. What would be the magnetic field at the center of a current carrying loop? Let us assume that the wire is a loop with a radius R and carries a current of I.

image courtesy of John Hopkins University
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Since r is always perpendicular to the direction of the current, we do not need to worry about messy integration.

Furthermore, since we are in a circular loop,

is equal to

. So we end up with

It is easily possible to find the magnetic field in many other geometries. The Biot-Savart Law is much, much, much more accurate than Ampere's Law (as its applications involve fewer assumptions). However, it is also much harder to apply. Therefore, it will tend to be the law used when Ampere's Law doesn't fit. For more practice, find other geometries of wires to practice with because nobody likes Biot-Savart.

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¹Details for the derivation of a magnetic field around a current carrying wire were provided courtesy of Carlos Valera, May 2011.