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Current Question (ID: 16589)

Question:

$\text{A series combination of } n_1 \text{ capacitors, each of value } C_1, \text{ is charged by a source of potential difference } 4 \text{ V.}$ $\text{When another parallel combination of } n_2 \text{ capacitors, each of value } C_2, \text{ is charged by a source of potential difference } V, \text{ it has the same (total) energy stored in it as the first combination has.}$ $\text{The value of } C_2 \text{ in terms of } C_1 \text{ is:}$

Options:

1. $\frac{2C_1}{n_1n_2}$
2. $16 \frac{n_2}{n_1} C_1$
3. $2 \frac{n_2}{n_1} C_1$
4. $\frac{16C_1}{n_1n_2}$

Solution:

$\text{Case I: When the capacitors are joined in series:}$ $U_{\text{series}} = \frac{1}{2} \frac{C_1}{n_1} (4V)^2$ $\text{Case II: When the capacitors are joined in parallel:}$ $U_{\text{parallel}} = \frac{1}{2} (n_2C_2)V^2$ $\text{Given, } U_{\text{series}} = U_{\text{parallel}}$ $\text{or } \frac{1}{2} \frac{C_1}{n_1} (4V)^2 = \frac{1}{2} (n_2C_2)V^2$ $\Rightarrow C_2 = \frac{16C_1}{n_2n_1}$

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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
}