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Current Question (ID: 20611)

Question:

$\text{A plane electromagnetic wave having a frequency } \nu = 23.9 \text{ GHz}$ $\text{propagates along the positive } z\text{-direction in free space. The peak}$ $\text{value of the Electric Field is } 60 \text{ V/m. Which among the following is}$ $\text{the acceptable magnetic field component in the electromagnetic}$ $\text{wave?}$

Options:

1. $\vec{B} = 2 \times 10^{-7} \sin(1.5 \times 10^2 x + 0.5 \times 10^{11} t) \hat{j}$
2. $\vec{B} = 60 \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t) \hat{k}$
3. $\vec{B} = 2 \times 10^{-7} \sin(0.5 \times 10^3 z + 1.5 \times 10^{11} t) \hat{j}$
4. $\vec{B} = 2 \times 10^{-7} \sin(0.5 \times 10^3 z + 1.5 \times 10^{11} t) \hat{i}$

Solution:

$\text{Hint: } \frac{E_0}{B_0} = c$ $\nu = 23.9 \text{ GHz}$ $E_0 = 60 \text{ V/m}$ $\therefore \frac{E_0}{B_0} = c = \frac{E_0}{C} = B_0 = \frac{E_0}{C} = \frac{60}{3 \times 10^8} = 2 \times 10^{-7}$ $\text{Since the wave is propagating in positive } z\text{-direction}$ $\text{So acceptable magnetic field component will be}$ $\vec{B} = 2 \times 10^{-7} \sin(0.5 \times 10^3 z + 1.5 \times 10^{11} t) \hat{j}$

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}