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 epsilon is always on the opposite side of the fraction with mu 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,

• or

• if you call the angle between r and I to be q.

That wasn’t so hard, was it? You might want to take a breather before continuing. Rested? Then let's prove the Biot-Savart Law.

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 find the proper equation to use . 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 (ain’t physics great?), so . R is the distance from the point to the wire, and r is the distance from the point-current element to the point. If is the distance from the point-current element to the closest point of the wire to the point, then . Furthermore, by the laws of trigonometry, sin θ = /r, so we end up with .

Isn't this messy? Well, here is where mathematics comes in handy. It turns out that

 (thank goodness for definite integrals!) so ,

which we already knew (there is another proof in your black book, p 807).

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. So . 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 2pr. 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.

Contributed by Lawrence Camarota, August 2004