In the figure, find out the magnetic field at $B$ (Given $I =2.5 \;A,r =5\, cm )$
AIIMS 2019, Diffcult
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The magnetic field at $B$ is given by,
$\vec{B}_{ B }=\vec{B}_{\text {wire }}+\vec{B}_{\text {ring }}$
$=\left[\frac{\mu_{0} i}{2 r}(-\hat{k})+\frac{\mu_{0} i}{4 \pi r}(-\hat{k})\right]+\left[\frac{\mu_{0} i}{2 \pi r}(-\hat{k})+\frac{\mu_{0} i}{2 r}(-\hat{k})\right]$
$=\frac{\mu_{0} i}{2 \pi r}(-\hat{k})+\left[\frac{\mu_{0} i}{2 \pi r}+\frac{\mu_{0} i}{2 r}\right](-\hat{k})$
$=\frac{\mu_{0} i}{2 r}\left[\frac{1}{\pi}+1\right]$ $....(I)$
Substitute $4 \pi \times 10^{-7}$ for $\mu_{0}, 2.5$ for $i$ and $5 \times 10^{-2}$ for $r$ in equation $(I).$
$\vec{B}_{ B }=\frac{\left(4 \pi \times 10^{-7}\right) 2.5}{2\left(5 \times 10^{-2}\right)}\left[\frac{1}{\pi}+1\right]$
$=10 \pi \times 10^{-6}\left[\frac{1}{\pi}+1\right]$
$=\pi \times\left[\frac{1}{\pi}+1\right] \times 10^{-5} T$
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