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Current Question (ID: 17425)

Question:

$\text{The electric field associated with an electromagnetic wave in vacuum is given by } E = 40 \cos(kz - 6 \times 10^8 t), \text{ where } E, z, \text{ and } t \text{ are in volt/m, meter, and second respectively. The value of the wave vector } k \text{ would be:}$

Options:

1. $2 \text{ m}^{-1}$
2. $0.5 \text{ m}^{-1}$
3. $6 \text{ m}^{-1}$
4. $3 \text{ m}^{-1}$

Solution:

$\text{(1) Electromagnetic wave equation:}$ $E = E_0 \cos(kz - \omega t) \quad \text{...(i)}$ $\text{Speed of electromagnetic wave, } v = \frac{\omega}{k}$ $\text{Given equation:}$ $E = 40 \cos(kz - 6 \times 10^8 t) \quad \text{...(ii)}$ $\text{Comparing Eqs. (i) and (ii), we get:}$ $\omega = 6 \times 10^8$ $\text{and } E_0 = 40$ $\text{Here, wave factor } k = \frac{\omega}{v} = \frac{6 \times 10^8}{3 \times 10^8} = 2 \text{ m}^{-1}$

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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
}