Electromagnetism / Electrodynamics I · Magnetostatics and vector potential
Helmholtz spacing cancels the axial curvature
Problem
Two identical coaxial circular loops of radius $a$ are centered at $z=\pm s/2$. Each carries current $I$ in the same sense, producing a field along $+z$ at the midpoint. Starting from the single-loop axis field $B_z(z)=\mu_0Ia^2/[2(a^2+z^2)^{3/2}]$, derive the pair field, expand it about $z=0$ through the quadratic term, and determine the separation $s$ that makes the quadratic coefficient vanish. Give the resulting midpoint field.
Hint
Write the two denominators using $z-s/2$ and $z+s/2$.
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