Import Question JSON

Current Question (ID: 20610)

Question:

$\text{An electromagnetic wave is represented by an electric field}$ $\vec{E} = E_0 \hat{n} \sin[\omega t + (\hat{s}y - 8z)].$ $\text{Taking unit factors in } x, y, \text{ and } z$ $\text{directions to be } \hat{i}, \hat{j}, \hat{k} \text{ the direction of propagation } \hat{s} \text{ is:}$

Options:

1. $\hat{s} = \frac{-4\hat{k} + 3\hat{j}}{5}$
2. $\hat{s} = \frac{4\hat{j} + 3\hat{k}}{5}$
3. $\hat{s} = \frac{-3\hat{j} + 4\hat{k}}{5}$
4. $\hat{s} = \frac{3\hat{i} + 4\hat{j}}{5}$

Solution:

$\text{Hint: The direction of propagation will be perpendicular to the electric field.}$ $\vec{E} = E_0 \hat{n} \left[ \omega t + \left( 6y - 8z \right) \right]$ $\vec{E} = E_0 \hat{n} \left[ \omega t - \left( 8z - 6y \right) \right]$ $\vec{E} = E_0 \hat{n} \left[ \omega t + \left( \frac{8}{10} \hat{k} - \frac{6}{10} \hat{j} \right) \cdot 10 \right]$ $\vec{E} = E_0 \hat{n} \left[ \omega t - \hat{s} \hat{k} \right]$ $\hat{s} = \text{direction of propagation}$ $\hat{s} = \left( \frac{8\hat{k} - 6\hat{j}}{10} \right)$ $= \frac{4\hat{k} - 3\hat{j}}{5}$ $= \frac{-3\hat{j} + 4\hat{k}}{5}$

Import JSON File

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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
}