Finding the magnitude of a magnetic field from a square loop

[Patdon10]

Homework Statement

A square loop, with sides of length L, carries current i. Find the magnitude of the magnetic field from the loop at the center of the loop, as a function of i and L. (Use any variable or symbol stated above along with the following as necessary: μ0.)

Homework Equations

magnetic field outside of a conductor:
u_0*I*r/A

The Attempt at a Solution

I got u_0*I*L/L^2

Not really sure what I should be doing differently? If it was in the shape of a circle it'd be easy, but because it's in a square, it's harder.

 

[epsilonjon]

I think the easiest way would be to try to solve the problem of finding the magnetic field produced by a straight current-carrying piece of wire of length L. Try to find the field at a point a distance x from the wire on its perpendicular bisector. You can do this by splitting the wire into infinitessimal lengths dl and then use the Biot and Savart law to calculate the field produced by dl. Then integrate along the length of the wire to find the total.

I think you should get

B = \frac{\mu_{0}I}{4\pi}\frac{L}{x\sqrt{x^{2}+(L/2)^{2} }}.
Now you've done the hard part it's just a matter of adding the fields from each of the 4 sides of the loop.

 

[Patdon10]

It seems pretty confusing, but I'll try it out and see what happens.

 

[epsilonjon]

It seems pretty confusing, but I'll try it out and see what happens.

I think that is the easiest way to do it. If you are trying to find the field from shapes like this then presumably you've covered the field from a straight current-carrying wire?

If it was in the shape of a circle it'd be easy, but because it's in a square, it's harder.

How would you do it for a circle?