# Magnetic field of a circular loop

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1. Nov 30, 2015

### shinobi20

1. The problem statement, all variables and given/known data
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 Br and Bθ 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.

2. Relevant equations
A(r) = k∫ (J(r') dτ') / |r-r'| = kI ∫ dr' / |r-r'| where k is μo/4π and I is the current

3. 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 Aφ

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

Bφ = 0
Br = - 1/r ∂/∂cosθ (Aφsinθ)
Bθ = - 1/r ∂/∂r (rAφ)

Is this correct?

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

2. Dec 1, 2015

### shinobi20

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

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