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GATE 2026 EE – Question 51

Electromagnetic Fields · Biot-Savart law, Ampere law, Curl, Faraday law, Lorentz force · 2 marks · Multiple choice

The figure shows an arbitrarily shaped planar conducting loop A in the XY plane. Two nonintersecting regions with areas $a_1$ and $a_2$ within the loop are subjected to magnetic fields $\vec B_1=\frac{m}{\sqrt2}\sin(\omega t)\,(1\,\hat x+0\,\hat y+1\,\hat z)$, and $\vec B_2=-\frac{n}{\sqrt2}\cos(2\omega t+\pi/4)\,(0\,\hat x+1\,\hat y+1\,\hat z)$, respectively.

What is the expression for the induced rms voltage in loop A?

Diagram for GATE 2026 EE question 51
  1. $\sqrt{\dfrac{a_1^2\omega^2m^2+4a_2^2\omega^2n^2}{4}}$
  2. $\sqrt{\dfrac{a_1^2\omega^2m^2+4a_2^2\omega^2n^2}{2}}$
  3. $\sqrt{\dfrac{a_1^2\omega^2m^2-2a_2^2\omega^2n^2}{2}}$
  4. $\sqrt{a_1^2\omega^2m^2+2a_2^2\omega^2n^2}$

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Correct answer: (A) $\sqrt{\dfrac{a_1^2\omega^2m^2+4a_2^2\omega^2n^2}{4}}$

Explanation

Only the $z$-components link the loop in the XY plane: $\phi_1=\dfrac{m}{\sqrt2}a_1\sin\omega t$ and $\phi_2=-\dfrac{n}{\sqrt2}a_2\cos(2\omega t+\pi/4)$. The emf amplitudes are $\dfrac{ma_1\omega}{\sqrt2}$ (at $\omega$) and $\dfrac{2na_2\omega}{\sqrt2}$ (at $2\omega$), so the rms values are $\dfrac{ma_1\omega}{2}$ and $na_2\omega$. Voltages of different frequencies add in power: $V_{rms}=\sqrt{\dfrac{a_1^2m^2\omega^2}{4}+a_2^2n^2\omega^2}=\sqrt{\dfrac{a_1^2\omega^2m^2+4a_2^2\omega^2n^2}{4}}$.