Two circular coils $P$ and $Q$of $100$ turns each have same radius of $\pi \mathrm{cm}$. The currents in $\mathrm{P}$ and $\mathrm{R}$ are $1 \mathrm{~A}$ and $2 \mathrm{~A}$ respectively. $\mathrm{P}$ and $\mathrm{Q}$ are placed with their planes mutually perpendicular with their centers coincide. The resultant magnetic field induction at the center of the coils is $\sqrt{\mathrm{x}} \mathrm{mT}$, where X=___.
$\left[\text { Use } \mu_0=4 \pi \times 10^{-7} \mathrm{TmA}^{-1}\right]$
JEE MAIN 2024, Diffcult
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$\mathrm{B}_{\mathrm{p}}=\frac{\mu_0 \mathrm{Ni}_1}{2 \mathrm{r}}=\frac{\mu_0 \times 1 \times 100}{2 \pi}=2 \times 10^{-3} \mathrm{~T}$
$\mathrm{~B}_{\mathrm{Q}}=\frac{\mu_0 \mathrm{Ni}_2}{2 \mathrm{r}}=\frac{\mu_0 \times 2 \times 100}{2 \pi}=4 \times 10^{-3} \mathrm{~T}$
$\mathrm{~B}_{\text {net }}=\sqrt{\mathrm{B}_{\mathrm{p}}^2+\mathrm{B}_{\mathrm{Q}}^2}$
$=\sqrt{20} \mathrm{mT}$
$\mathrm{x}=20$
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