Import Question JSON

Current Question (ID: 20107)

Question:

$\text{Three concentric metal shells } A, B \text{ and } C \text{ of respective radii } a, b \text{ and } c \,(a < b < c) \text{ have surface charge densities } +\sigma, -\sigma \text{ and } +\sigma \text{ respectively. The potential of the shell } B \text{ is:}$

Options:

1. $\frac{\sigma}{\varepsilon_0} \left[ \frac{a^2 - b^2}{a} + c \right]$
2. $\frac{\sigma}{\varepsilon_0} \left[ \frac{a^2 - b^2}{b} + c \right]$
3. $\frac{\sigma}{\varepsilon_0} \left[ \frac{b^2 - c^2}{b} + a \right]$
4. $\frac{\sigma}{\varepsilon_0} \left[ \frac{b^2 - c^2}{c} + a \right]$

Solution:

$\text{Hint: } V = \frac{kQ}{R}$ $V_B = \frac{kq_A}{a} + \frac{kq_B}{b} + \frac{kq_C}{c}$ $= \frac{1}{4\pi\varepsilon_0} \frac{\sigma 4\pi a^2}{b} + \frac{1}{4\pi\varepsilon_0} \frac{(-\sigma) 4\pi b^2}{b} + \frac{1}{4\pi\varepsilon_0} \frac{\sigma \times 4\pi c^2}{c}$ $= \frac{\sigma}{\varepsilon_0} \left( \frac{a^2 - b^2}{b} + c \right)$

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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
}