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 \(\mathrm{x}=\) _______. \(\left[\text { Use } \mu_0=4 \pi \times 10^{-7} \mathrm{TmA}^{-1}\right]\)

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Answer & Solution

Correct Answer

(B) \(20\)

Detailed explanation

\(\mathrm{B}_{\mathrm{p}}=\frac{\mu_0 \mathrm{Ni}_1}{2 \mathrm{r}}=\frac{\mu_0 \times 1 \times 100}{2 \pi}=2 \times 10^{-3} \mathrm{~T}\)

\(\mathrm{~B}_{\mathrm{Q}}=\frac{\mu_0 \mathrm{Ni}_2}{2 \mathrm{r}}=\frac{\mu_0 \times 2 \times 100}{2 \pi}=4 \times 10^{-3} \mathrm{~T}\)

\(\mathrm{~B}_{\text {net }}=\sqrt{\mathrm{B}_{\mathrm{p}}^2+\mathrm{B}_{\mathrm{Q}}^2}\)

\(=\sqrt{20} \mathrm{mT}\)

\(\mathrm{x}=20\)

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