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 with 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 IdlπR since the total current I flows through the semicircular cross section of perimeter πR. Putting dB1=dB2=dB we have
dB=μIdlπR2πR=μ0IRdθπR2πR=μ0Idθ2π2R
The X-component of the above field is μ0I2π2Rsinθdθ
The field due to the entire conductor is B=∫π0[μ0I2π2Rsinθ]dθ
or, B=μ0Iπ2R since ∫π0sinθdθ=2