Two identical long current-carrying wires are bent into the two shapes shown in the figure. The semicircular arcs in both cases have radius r and carry the same current I. If the magnetic-field magnitudes at the centres P and Q are B₁ and B₂, respectively, what is B₁ / B₂?

Question

G1 Physics · Accepted Answer

For a semicircular arc of radius r, B(arc) = μ₀I ÷ (4r). For a semi-infinite straight wire at perpendicular distance r, B(semi-infinite) = μ₀I ÷ (4πr). From configuration I in the source figure, two straight-wire contributions act in the same direction as the semicircular contribution. Hence, B₁ = μ₀I ÷ (4r) + 2[μ₀I ÷ (4πr)]. Factor μ₀I ÷ (4r): B₁ = [μ₀I ÷ (4r)][1+2 ÷ π]. For configuration II, there is one relevant straight-wire contribution: B₂ = μ₀I ÷ (4r) + μ₀I ÷ (4πr) = [μ₀I ÷ (4r)][1+1 ÷ π…