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JEE Main 2020 (9 Jan, Shift 2) · Nature of EM Waves & Wave Equation

Nature of EM Waves & Wave Equation question from JEE Main 2020 (9 Jan, Shift 2)

Q58JEE Main 2020 · 9 JAN · SHIFT 2Difficult

Nature of EM Waves & Wave Equation

A plane electromagnetic wave is propagating along the direction i^+j^2\frac{\hat{i} + \hat{j}}{\sqrt{2}}, with its polarization along the direction k^\hat{k}. The correct form of the magnetic field of the wave would be (here B0 is an appropriate constant):

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Answer: (a)

xyn (i+j)B (i−j)E along k
  1. E⃗×B⃗\vec{E} \times \vec{B} points along propagation, so B^=n^×E^\hat{B} = \hat{n} \times \hat{E}.
  2. B^=i^+j^2×k^=−j^+i^2=i^−j^2\hat{B} = \dfrac{\hat{i} + \hat{j}}{\sqrt{2}} \times \hat{k} = \dfrac{-\hat{j} + \hat{i}}{\sqrt{2}} = \dfrac{\hat{i} - \hat{j}}{\sqrt{2}}.
  3. A wave moving along +n^+\hat{n} has phase (ωt−k n^⋅r⃗)(\omega t - k\,\hat{n}\cdot\vec{r}).
  4. So the answer is (a) B0i^−j^2cos⁡(ωt−ki^+j^2)B_0 \dfrac{\hat{i} - \hat{j}}{\sqrt{2}} \cos\left(\omega t - k\dfrac{\hat{i} + \hat{j}}{\sqrt{2}}\right).
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