Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson
Physics — E&M · toroidal-loop-self-inductance
Problem
A circular loop of mean radius a is made of wire having a circular cross section of radius b, with b≪a. The sketch shows the relevant dimensions and coordinates for this problem.
(a) Using (5.37), the expression for the vector potential of a filamentary circular loop, and appropriate approximations for the elliptic integrals, show that the vector potential at the point P near the wire is approximately
Aϕ=(μ0I/2π)[ln(8a/ρ)−2]
where ρ is the transverse coordinate shown in the figure and corrections are of order (ρ/a)cosϕ and (ρ/a)2.
(b) Since the vector potential of part a is, apart from a constant, just that outside a straight circular wire carrying a current I, determine the vector potential inside the wire (ρ<b) in the same approximation by requiring continuity of Aϕ and its radial derivative at ρ=b, assuming that the current is uniform in density inside the wire:
Aϕ=(μ0I/4π)(1−ρ2/b2)+(μ0I/2π)[ln(8a/b)−2],ρ<b
(c) Use (5.149) to find the magnetic energy, hence the self-inductance,
L=μ0a[ln(8a/b)−7/4]
Are the corrections of order b/a or (b/a)2? What is the change in L if the current is assumed to flow only on the surface of the wire (as occurs at high frequencies when the skin depth is small compared to b)?