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Current Question (ID: 17441)

Question:

$\text{The electric field of an electromagnetic wave in free space is given by}$ $\vec{E} = 10 \cos(10^7 t + kx) \hat{j} \ \text{V/m, where} \ t \ \text{and} \ x \ \text{are in seconds and meters respectively. It can be inferred that:}$ $1. \ \text{The wavelength} \ \lambda \ \text{is} \ 188.4 \ \text{m.}$ $2. \ \text{The wave number} \ k \ \text{is} \ 0.33 \ \text{rad/m.}$ $3. \ \text{The wave amplitude is} \ 10 \ \text{V/m.}$ $4. \ \text{The wave is propagating along} \ +x \ \text{direction}$ $\text{Which one of the following pairs of statements is correct?}$

Options:

1. $(3) \ \text{and} \ (4)$
2. $(1) \ \text{and} \ (2)$
3. $(2) \ \text{and} \ (3)$
4. $(1) \ \text{and} \ (3)$

Solution:

$\text{The electric field of electromagnetic wave:}$ $\vec{E} = 10 \cos(10^7 t \pm kx) \hat{j}$ $\text{Amplitude} = 10 \ \text{V/m}$ $\therefore \ c = \frac{\omega}{k}$ $\therefore \ 3 \times 10^8 = \frac{10^7}{k}$ $\text{or} \ k = \frac{1}{30}$ $\text{or} \ \frac{2\pi}{\lambda} = \frac{1}{30}$ $\text{or} \ \lambda = 188.4 \ \text{m}$ $\text{So, (1) and (3) statements are correct.}$

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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
}