1. The problem statement, all variables and given/known data 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? 2. Relevant equations [tex] d{\mathbf{B}} = \frac{{\mu _0 }}{{4\pi }}\frac{{Id{\mathbf{s}} \times {\mathbf{\hat r}}}}{{r^2 }} [/tex] 3. 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 [tex]\mathbf{\hat r}[/tex] 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 [tex]ds\neq rd\phi[/tex]. 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.
First thing you want to do is draw a picture and realise this is a highly symmetric problem. From symmetry you can determine the direction of [itex]\boldsymbol{B}[/itex], draw this vector. Then draw the x and y components. Due to symmetry can you tell what the magnitude of [itex]B_y[/itex] will be? Can you express [itex]B_x[/itex] in terms of B? Why do you think the angle between [itex]d\boldsymbol{s}[/itex] and [itex]\mathbf{\hat r}[/itex] changes? It does not. So [itex]d\boldsymbol{s} \times \mathbf{\hat r}=ds[/itex]. 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 [itex]\phi[/itex]? Try to enter all information into the Bio Savart law now.