JEE Advanced 2023 Paper 2 Q07 Physics Waves & Optics Wave Optics Hard

JEE Advanced 2023 Paper 2 · Q07 · Wave Optics

The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by $\vec{E} = 30(2\hat{x} + \hat{y})\sin\left[2\pi\left(5\times 10^{14}t - \dfrac{10^7 z}{3}\right)\right]$ V·m$^{-1}$. Which of the following option(s) is(are) correct? [Given: $c = 3\times 10^8$ m·s$^{-1}$]

  1. A. $B_x = -2\times 10^{-7}\sin\left[2\pi\left(5\times 10^{14}t - \dfrac{10^7 z}{3}\right)\right]$ Wb·m$^{-2}$
  2. B. $B_y = 2\times 10^{-7}\sin\left[2\pi\left(5\times 10^{14}t - \dfrac{10^7 z}{3}\right)\right]$ Wb·m$^{-2}$
  3. C. The wave is polarized in the $xy$-plane with polarization angle $30°$ with respect to the $x$-axis.
  4. D. The refractive index of the medium is $2$.
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:A, D

Solution

Wave propagates in $+z$. From $\omega = 2\pi\cdot 5\times 10^{14}$ and $k = 2\pi\cdot 10^7/3$: $v = \omega/k = 1.5\times 10^8$ m/s. Refractive index $n = c/v = 2$ — (D) TRUE. Polarization: $\vec{E}$ has components $(2,1,0)$, so polarization angle from $x$-axis is $\tan^{-1}(1/2) \approx 26.57°$, NOT $30°$ — (C) FALSE. Magnetic field: $\vec{B} = (\hat{k}\times\vec{E})/v$ where $\hat{k} = \hat{z}$. $\hat{z}\times(2\hat{x}+\hat{y}) = 2\hat{y} - \hat{x}$. Amplitude $E_0/v = 30\sqrt{5}/(1.5\times 10^8)$. Component magnitudes: $B_{0x} = -30/(1.5\times 10^8) = -2\times 10^{-7}$ — (A) TRUE. $B_{0y} = 60/(1.5\times 10^8) = 4\times 10^{-7}$, NOT $2\times 10^{-7}$ — (B) FALSE.

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