An infinite sequence of concentric semicircular wires carries current I as shown. Their radii obey Rₙ=αⁿ⁻¹R₁, α>1. The net field at the common centre is [μ₀I/(4R₁)]·[α/(β+α)]. Find the numerical constant β.

Question

G1 Physics · Accepted Answer

Each semicircle contributes μ₀I/(4Rₙ), with alternating sign because successive semicircles carry current in opposite senses. Thus B=[μ₀I/(4R₁)](1-1/α+1/α²-⋯)=[μ₀I/(4R₁)]·α/(α+1). Comparing with α/(α+β) gives β=1.

An infinite sequence of concentric semicircular wires carries current I as shown. Their radii obey Rₙ=αⁿ⁻¹R₁, α&gt;1. The net field at the common centre is [μ₀I/(4R₁)]·[α/(β+α)]. Find the numerical constant β.

An infinite sequence of concentric semicircular wires carries current I as shown. Their radii obey Rₙ=αⁿ⁻¹R₁, α>1. The net field at the common centre is [μ₀I/(4R₁)]·[α/(β+α)]. Find the numerical constant β.