Explanation

(d) Consider the wire to be made up of large number of thin wires of infinite length. Consider such wire of thickness $dl$ subtending an angle $d\theta$ at centre.
Current through this wire,
$dl = \frac{d\theta}{\pi}\ I,$
$\therefore$ Magnetic field at centre due to this portion.
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$dB = \frac{\mu_{0}}{4\pi}\ .\frac{2dI}{R}$
$\ \ \ = \frac{\mu_{0}I}{2\pi^{2}R}\ d\theta$
Net magnetic field at the centre.
$B = \ \ \int_{- \pi/2}^{\pi/2}{dB}\cos\theta = \frac{\mu_{0}I}{2\pi^{2}R}$
$\ \ \ \ \ \ \ \ \ \ \int_{- \pi/2}^{\pi/2}{\cos\theta}d\theta = \frac{\mu_{0}I}{\pi^{2}R}$