How to Find Magnetic Field Components of a Circular Loop?

[shinobi20]

Homework Statement

A circular loop of radius R is on the xy plane and the center is at the origin, the current is flowing in a counter-clockwise manner. a) Let Q=(a,0,a) be a point such that a>>R. Find B_r and B_θ at Q. b) Let Q=(ha,0,0) be a point on the x-axis such that h<1. Find the vector potential A at Q as a power series of h.

Homework Equations

A(r) = k∫ (J(r') dτ') / |r-r'| = kI ∫ dr' / |r-r'| where k is μ_o/4π and I is the current

The Attempt at a Solution

a) From azimuthal symmetry, we can restrict the situation to points r on the xz plane.
dr'=(dx', dy', 0)=(-Rsinφ', Rcosφ', 0)dφ'. Since the only non vanishing component of A is A_φ

A_φ(r) = kI ∫ (Rcosφ' dφ') / |r-r'| from 0 to 2π

B_φ = 0
B_r = - 1/r ∂/∂cosθ (A_φsinθ)
B_θ = - 1/r ∂/∂r (rA_φ)

Is this correct?

For part b) I don't know if it is a multipole expansion or somethin' else... Any help?

 

[shinobi20]

I think for part b) it can be expressed as a multipole expansion, A_φ = kI ∑_l=0 ( ha_< P_l(0) P_l(cosθ) )/( l(l+1) ). Is this correct?