The magnetic field at the centre O of the arc shown in the figure is

Magnetic Field Vector: The magnetic field is described mathematically as a vector field . This vector field can be plotted directly as a set of many vectors drawn on a grid. Each vector points in the direction that a compass would point and has length dependent on the strength of the magnetic force.
Magnetic Field Lines: An alternative way to represent the information contained within a vector field is with the use of field lines . Here we dispense with the grid pattern and connect the vectors with smooth lines.

Here, $a =\frac{r}{\sqrt{2}}$ Magnetic field at point O due to AB is $B_{1}=\frac{\mu_{0}}{4\pi } \frac{I}{a}=\frac{\mu_{0} I}{4\pi\left(r /\sqrt{2}\right)}$ Magnetic field at point O due to BCD is $B_{2}=\frac{\mu_{0}I}{4\pi r} \left(\frac{\pi}{2}\right)$ Magnetic field at point O due to DE is $B_{3}=\frac{\mu_{0}I}{4\pi a}=\frac{\mu_{0}I}{4\pi\left(r /\sqrt{2}\right)}$ Resultant magnetic field at point O is $B = B_{1} + B_{2} + B_{3}$ $=\frac{\mu_{0}I}{4\pi\left(r /\sqrt{2}\right)}+\frac{\mu_{0}I}{4\pi r} \left(\frac{\pi}{2}\right)+\frac{\mu_{0}I}{4\pi\left(r /\sqrt{2}\right)}$ $=\frac{\mu_{0}I}{4\pi r}\left(\sqrt{2}+\frac{\pi}{2}+\sqrt{2}\right)=\frac{\mu_{0}I}{4\pi r}\left(2\sqrt{2}+\frac{\pi}{2}\right)$ $=\frac{\mu_{0}2I}{4\pi r}\left(\sqrt{2}+\frac{\pi}{4}\right)$ $=\frac{10^{-7}\times2I}{r}\left(\sqrt{2}+\frac{\pi}{4}\right)\quad\quad\quad\left[\because \frac{\mu_{0}}{4\pi}=10^{-7}\right]$

$2I\left(\sqrt{2}+\pi\right)\times\frac{10^{-7}}{r}$

$2I\left(\sqrt{2}+\frac{\pi}{4}\right)\times\frac{10^{-7}}{r}$

$I\left(\sqrt{2}+\pi\right)\times\frac{10^{-7}}{r}$

$I\left(\sqrt{2}+\frac{\pi}{4}\right)\times\frac{10^{-7}}{r}$