A plane current spiral is described by r=aθ for θ₀≤θ≤2π, where θ₀>0, and carries current I. What is the magnetic field at the pole due only to the spiral portion?
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A plane current spiral is described by r=aθ for θ₀≤θ≤2π, where θ₀>0, and carries current I. What is the magnetic field at the pole due only to the spiral portion?

G1 Physics · Accepted Answer

For a polar spiral element, the component relevant at the pole gives dB=μ₀I dθ/(4πr)=μ₀I dθ/(4πaθ). Integration from θ₀ to 2π gives the logarithmic result. A non-zero θ₀ avoids the singular idealization at r=0.