A current I flows in an infinitely long wire with cross-section in the form of a semicircular ring of radius R. The magnitude of the magnetic induction at its axis is

In the figure we have shown the cross section (of the given wire) lying in the XY plane. The length of the wire is along the Z-axis and the current in the wire is supposed to flow along the negative Z-direction. The broad infinitely long wire can be imagined to be made of a large number of infinitely long straight wire strips, each of small width dl. With reference to the figure, we have dl = Rdθ. The magnetic flux density due to the above strip is shown as dB₁ in the figure. It has an X-component dB₁sinθ and Y-component dB₁cosθ. When we consider a similar strip of the same width dl located symmetrically with respect to the Y-axis, we obtain a contribution dB₂ to the flux density. The flux density dB₂ has the same magnitude as dB₁. It has X-component dB₂sinθ and Y-component dB₂cosθ. The X-components of dB₁ and dB₁ are in the same magnitude and direction and they add up. But the Y-components of dB₁ and dB₁ are in opposite directions and have the same magnitude. Therefore they get canceled. The entire conductor therefore produces a resultant magnetic field along the negative X-direction. The wire strip of width dl can be imagined to be an ordinary thin straight infinitely long wire carrying current Idl/πR since the total current I flows through the semicircular cross section of perimeter πR. Putting dB₁ = dB₂ = dB we have dB = µ₀Idl/(2πR) = µ₀IRdθ/(2πR) = µ₀Idθ/(2πR) The X-component of the above field is µ₀I/(2π2R)sinθdθ The field due to the entire conductor is B = ∫₀^π[µ₀I/(2π2R)sinθ]dθ or, B = µ₀I/2R since ∫₀^πsinθdθ = 2