Hi again,
Here's an interesting way to prove what the B at the center of the loop is...
First calculate the value of B at a distance R from the center of a finite wire segment of length L using Biot-Savart, call that B1.
Next, since we know that the circumference of a circle is 2*pi*r and the loop follows this path, we place one length L of a wire above right at some point on this circle. The wire length is to be placed on the circle at one point on the circle and tangent to the circle at that point.
Now since we know what B1 is from above and we know the length L of the short wire placed on the circle circumference, we know the value of B at the center right now is B1 also. That's using one wire segment.
But the circle is not complete yet because we've only used one short length of wire, so what we have to do next is place more wire segments of the same length next to each other so that they form a circle, a crude circle but the wire segments follow the circumference and enclose the same area as the circle once we have enough wire segments.
To express this process mathematically, all we have to do is make each length equal to the circumference divided by some number N where N will be the number of segments used to approximate the circle:
L=2*pi*r/N
So now we have the length of each segment, but since we want to go all the way around the circle that means we have to place N such segments around the circle. This means when we do this we also multiply the B of one segment by N and we can do this because of superposition. What we get at the center then is B1*N.
So what we would end up with using Biot-Savart is a function in R and L where B=F(R,L), F being this function for one segment and L each length, and L is defined above. Then we multiply this function by N (because of the N segments) and then take the limit as we let N approach infinity, and what we get is:
B=u0*I/(2*R)
If you would like to try this yourself as an exercise you'll have to start with Biot-Savart, but in case that's a little too much work here is a function you can use:
B=L*I*u0/(2*pi*R*sqrt(4*R^2+L^2))
[LATEX]B=\frac{L I \mu_0}{2 \pi R \sqrt{4 R^2+L^2}}[/LATEX]
That is a function that describes the field at the mid point of the current segment wire length where the wire length is L and the distance from the wire is R. That can be used in this proof and was derived directly from Biot-Savart. See attached diagram.
When you are done you might want to multiply by N but this N is the number of turns in the coil. The N above is the number of wire segments.