Q122 / 150 · MHT CET 2023 (12 May Shift 1) · Physics

Two similar coils each of radius $\mathrm{R}$ are lying concentrically with their planes at right angles to each other. The current flowing in them are I and 2I. The resultant magnetic field of induction at the centre will be $\left(\mu_0=\right.$ Permeability of vacuum) Figure

  • A
    $\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}$
  • B
    $\frac{\mu_0 \mathrm{I}}{\mathrm{R}}$
  • C
    $\frac{3 \mu_0 \mathrm{I}}{2 \mathrm{R}}$
  • Correct answer
    D
    $\frac{\sqrt{5} \mu_0 \mathrm{I}}{2 \mathrm{R}}$

Solution

$$ \mathrm{B}_1=\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}, \mathrm{B}_2=\frac{\mu_0(2 \mathrm{I})}{2 \mathrm{R}} $$ Resultant magnetic field, $B=\sqrt{\mathrm{B}_1+\mathrm{B}_2}$ $$ \begin{array}{ll} \therefore & B=\sqrt{\left(\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}\right)^2+\left(\frac{\mu_0(2 \mathrm{I})}{2 \mathrm{R}}\right)^2} \\ \therefore & \mathrm{B}=\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}} \sqrt{1^2+2^2} \\ \therefore & \mathrm{B}=\frac{\sqrt{5} \mu_0 \mathrm{I}}{2 \mathrm{R}} \end{array} $$

Hence, the correct answer is option: 'D'.

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