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Current Question (ID: 16596)

Question:

$\text{Maximum charge stored on a metal sphere of radius 15 cm may be } 7.5 \, \mu\text{C} . \text{ The potential energy of the sphere in this case is:}$

Options:

1. $9.67 \, \text{J}$
2. $0.25 \, \text{J}$
3. $3.25 \, \text{J}$
4. $1.69 \, \text{J}$

Solution:

$\text{Capacitance of a spherical capacitor is } C = 4\pi\varepsilon_0 r.$ $\text{Step 1: Find the capacitance of the spherical capacitor.}$ $\text{The capacitance of a spherical capacitor is given by:}$ $C = 4\pi\varepsilon_0 r$ $\text{Step 2: Find the potential energy of the capacitor.}$ $\text{Using,}$ $U = \frac{Q^2}{2C}$ $\text{Step 3: Put the value of } Q \text{ and } C \text{ and solve for } U.$ $U = \frac{(7.5 \times 10^{-6})^2 \times 9 \times 10^9}{2 \times 15 \times 10^{-2}}$ $= 1.69 \, \text{J}$

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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
}