Import Question JSON

Current Question (ID: 16428)

Question:

$\text{A charged ball } B \text{ hangs from a silk thread } S, \text{ which makes an angle } \theta \text{ with a large charged conducting sheet } P, \text{ as shown in the figure. The surface charge density } \sigma \text{ of the sheet is proportional to:}$

Options:

1. $\sin \theta$
2. $\tan \theta$
3. $\cos \theta$
4. $\cot \theta$

Solution:

$\text{Hint: Balance the components of forces in the horizontal and vertical directions.}$ $\text{Step: Find the surface charge density } \sigma \text{ of the sheet}$ $\text{Since the charged ball will come to equilibrium at some inclination, balance the components of forces and hence find the proportionality of surface charge density with inclination.}$ $\text{Balance the forces in the horizontal and vertical directions.}$ $T \sin \theta = qE$ $T \cos \theta = mg$ $\text{Put the value of the electric field due to a uniformly charged infinite sheet.}$ $\Rightarrow \tan \theta = \frac{qE}{mg} = \frac{q}{mg} \times \frac{\sigma}{2\varepsilon_0}$ $\Rightarrow \sigma \propto \tan \theta$ $\text{Hence, option (2) is the correct answer.}$

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Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

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
}