Magnetic field of a circular loop of wire

[tjkubo]

Homework Statement

I know how to find the magnetic field at the center of a circular loop of wire carrying current. If the radius of the loop is R, how do you find the magnetic field at a distance a from the center of the loop where a<R?

Homework Equations

$$d{\mathbf{B}} = \frac{{\mu _0 }}{{4\pi }}\frac{{Id{\mathbf{s}} \times {\mathbf{\hat r}}}}{{r^2 }}$$

The Attempt at a Solution

The small current element ds is always tangent to the loop. r varies from R-a to R+a. The angle θ between ds and $$\mathbf{\hat r}$$ seems to vary from 90° to some maximum angle that depends on a.
Also, if you define ϕ to be the angle around P from the place you first start to integrate to ds, then $$ds\neq rd\phi$$.

This is as far as I can analyze. I have no idea what to do with the angles. I am guessing there is some kind of relationship between r and θ or between r and ϕ or between θ and ϕ that I can't see.

 

[Cyosis]

First thing you want to do is draw a picture and realize this is a highly symmetric problem. From symmetry you can determine the direction of $\boldsymbol{B}$, draw this vector. Then draw the x and y components. Due to symmetry can you tell what the magnitude of $B_y$ will be? Can you express $B_x$ in terms of B?

Why do you think the angle between $d\boldsymbol{s}$ and $\mathbf{\hat r}$ changes? It does not. So $d\boldsymbol{s} \times \mathbf{\hat r}=ds$.

How does the distance r from the loop to a depend on known variables and does the magnitude of B vary when you rotate over the angle $\phi$?

Try to enter all information into the Bio Savart law now.