Question

The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by $\vec{E}$=30(2𝑥̂ + 𝑦̂)sin [2𝜋 (5×10¹⁴𝑡 −$\frac{10^7}{3}z$)] Vm⁻¹. Which of the following option(s) is(are) correct? [Given: The speed of light in vacuum, 𝑐 = 3 × 10⁸ ms⁻¹]

• A. 𝐵_𝑥 = −2 × 10⁻⁷ sin [2𝜋(5×10¹⁴𝑡−$\frac{10^7}{3}z$)]Wb⁻²
• B. 𝐵_y=2×10⁻⁷ sin[2𝜋(5×10¹⁴𝑡−$\frac{10^7}{3}z$)]Wb⁻²
• C. The wave is polarized in the 𝑥𝑦-plane with a polarization angle 30° with respect to the 𝑥-axis.
• D. The refractive index of the medium is 2

Answer 1

The Correct Option is A, D

the speed of light in a medium is V=$\frac{w}{k}$
$v=\frac{3\times5\times10^{14}}{10^7}$
$v=1.5\times10^8$
refractive index = $\mu$ = $\frac{C}{V}=\frac{3\times10^8}{1.5\times10^8}=2$
$\mu=2$
given, $\vec{E}=30(2\hat{x}+\hat{y})\,sin(2\pi(5\times10^{14}-\frac{10^7}{3}))^\frac{1}{m}$

$B_0=\frac{E_0}{V}=\frac{30\sqrt{5}}{1.5\times10^8}$
Direction of $\vec{B_0}$ is ($\vec{V}\times\vec{E}$)
$\vec{V}\times\vec{E}=\hat{k}\times\frac{2\hat{i}+\hat{j}}{\sqrt5}$
$(\frac{-\hat{i}+2\hat{j}}{\sqrt5})$ put value $\vec{B_0}$ = $\frac{30\sqrt5}{1.5\times10^8}\times(\frac{-\hat{i}+2\hat{j}}{\sqrt5})$
$B_x=-2\times10^7$
 

Answer 2

 Explanation:
- The speed of light $V$ in a medium is given by $\frac{\omega}{k}$.
- Using the provided frequency and wave number, $v = \frac{3 \times 5 \times 10^{14}}{10^7}$.
- This calculates to $v = 1.5 \times 10^8$ m/s.
- The refractive index $\mu$ is calculated as $\frac{C}{V} = \frac{3 \times 10^8}{1.5 \times 10^8} = 2$.
- Given the electric field $\vec{E} = 30(2\hat{x}+\hat{y})\sin\left[2\pi\left(5\times10^{14}t-\frac{z}{3\times10^7}\right)\right] \, \text{Vm}^{-1}$.
- The magnetic field amplitude $B_0 = \frac{E_0}{V} = \frac{30\sqrt{5}}{1.5\times10^8}$.
- The direction of $\vec{B_0}$ is given by $\vec{V} \times \vec{E}$.
- Calculating $\vec{V} \times \vec{E} = \hat{k} \times \frac{2\hat{i} + \hat{j}}{\sqrt{5}} = \frac{-\hat{i} + 2\hat{j}}{\sqrt{5}}$.
- Therefore, $\vec{B_0} = \frac{30\sqrt{5}}{1.5\times10^8} \times \frac{-\hat{i}+2\hat{j}}{\sqrt{5}}$.
- Finally, $B_x = -2 \times 10^{-7}$.
Further:
- The speed of light $V$ in the medium is derived using $V = \frac{\omega}{k}$.
- Calculating with the given values, $V = 1.5 \times 10^8 \, \text{m/s}$.
- The refractive index $\mu$ is found to be 2 using $\mu = \frac{C}{V}$.
- Given the electric field expression, the magnetic field $B_0$ and its direction are determined by $\vec{V} \times \vec{E}$.
- The correct options are A and D.