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

(A) $B_{x}=-2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^{7}}{3} z\right)\right] \mathrm{Wbm}^{-2}$.; (D) The refractive index of the medium is 2.

(B) $B_{y}=2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^{7}}{3} z\right)\right] \mathrm{Wbm}^{-2}$

(C) The wave is polarized in the $xy$-plane with polarization angle $30^{\circ}$ with respect to the $x$-axis.

EM Waves — JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 Question with Solution

The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by

E = 30 (2 \overset{x}{^} + \overset{y}{^}) sin [2 π (5 \times 1 0^{14} t - \frac{1 0 ^{7}}{3} z)] V m^{- 1}

. Which of the following option(s) is(are) correct?
[Given: The speed of light in vacuum,

c = 3 \times 1 0^{8} m s^{- 1}

]

(A) $B_{x} = - 2 \times 1 0^{- 7} sin [2 π (5 \times 1 0^{14} t - \frac{1 0 ^{7}}{3} z)] Wbm^{- 2}$ .✓ Correct
(B) $B_{y} = 2 \times 1 0^{- 7} sin [2 π (5 \times 1 0^{14} t - \frac{1 0 ^{7}}{3} z)] Wbm^{- 2}$
(C)The wave is polarized in the $x y$ -plane with polarization angle $3 0^{\circ}$ with respect to the $x$ -axis.
(D)The refractive index of the medium is 2.✓ Correct

Answer: (A), (D)

Solution

Speed of wave in medium:

c_{medium} = \frac{ω}{k} = \frac{5 \times 1 0 ^{14}}{1 0 ^{7} /3} = 1.5 \times 1 0^{8}

m/s.

Refractive index:

μ = \frac{c}{c _{medium}} = \frac{3 \times 1 0 ^{8}}{1.5 \times 1 0 ^{8}} = 2

. ✓ (Option D)

Magnitude of magnetic field:

B = \frac{E}{c _{medium}} = \frac{30 5}{1.5 \times 1 0 ^{8}} = 25 \times 1 0^{- 7}

T.

Direction of

B

: Since wave propagates in

+ z

direction and

E \propto (2 \overset{x}{^} + \overset{y}{^})

, we have

B \propto \hat{k} \times (2 \hat{i} + \hat{j}) = 2 \hat{j} - \hat{i}

.

Therefore:

B = 2 \times 1 0^{- 7} (- \hat{i} + 2 \hat{j}) sin [...]

, giving

B_{x} = - 2 \times 1 0^{- 7} sin [...]

. ✓ (Option A)

Polarization angle:

tan θ = \frac{1}{2}

, so

θ \neq = 30°

. ✗ (Option C)

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · One or more correct options

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · EM Waves

Solution

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