\(\vec{B}_{ B }=\vec{B}_{\text {wire }}+\vec{B}_{\text {ring }}\)

\(=\left[\frac{\mu_{0} i}{2 r}(-\hat{k})+\frac{\mu_{0} i}{4 \pi r}(-\hat{k})\right]+\left[\frac{\mu_{0} i}{2 \pi r}(-\hat{k})+\frac{\mu_{0} i}{2 r}(-\hat{k})\right]\)

\(=\frac{\mu_{0} i}{2 \pi r}(-\hat{k})+\left[\frac{\mu_{0} i}{2 \pi r}+\frac{\mu_{0} i}{2 r}\right](-\hat{k})\)

\(=\frac{\mu_{0} i}{2 r}\left[\frac{1}{\pi}+1\right]\)      \(....(I)\)

Substitute \(4 \pi \times 10^{-7}\) for \(\mu_{0}, 2.5\) for \(i\) and \(5 \times 10^{-2}\) for \(r\) in equation \((I).\)

\(\vec{B}_{ B }=\frac{\left(4 \pi \times 10^{-7}\right) 2.5}{2\left(5 \times 10^{-2}\right)}\left[\frac{1}{\pi}+1\right]\)

\(=10 \pi \times 10^{-6}\left[\frac{1}{\pi}+1\right]\)

\(=\pi \times\left[\frac{1}{\pi}+1\right] \times 10^{-5} T\)