The magnitude of magnetic field at point ' \(O\) ' in the following figure will be

Magnetic field due to current carrying arc is,
\(B=\frac{\mu_0 I}{4 \pi r} \times \theta=\frac{\mu_0 I}{4 \pi r} \times \frac{\pi}{2}=\frac{\mu_0 I}{8 r}\)
Magnetic field due to semi-infinite current carrying straight wires \(A B\) and \(C D\) is,
\(\frac{2 \mu_0 I}{4 \pi r}=\frac{\mu_0 I}{2 \pi r}\)
\(\therefore \quad\) Total magnetic field at ' O ' will be,
\(\begin{aligned}
\frac{\mu_0 I}{2 \pi r}+\frac{\mu_0 I}{8 r} & =\frac{\mu_0 I}{2 r}\left(\frac{1}{\pi}+\frac{1}{4}\right) \\
& =\frac{\mu_0}{4 \pi} \frac{1}{r}\left(\frac{2 \pi}{\pi}+\frac{2 \pi}{4}\right) \\
& =\frac{\mu_0}{4 \pi} \frac{1}{r}\left(2+\frac{\pi}{2}\right)
\end{aligned}\)