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Current Question (ID: 20126)

Question:

$\text{A capacitor } C \text{ is fully charged with voltage } V_0. \text{ After disconnecting the voltage source, it is connected in parallel with another uncharged capacitor of capacitance } \frac{C}{2}. \text{ The energy loss in the process after the charge is distributed between the two capacitors is:}$

Options:

1. $\frac{1}{6} CV_0^2$
2. $\frac{1}{3} CV_0^2$
3. $\frac{1}{4} CV_0^2$
4. $\frac{1}{2} CV_0^2$

Solution:

$\text{Hint: } U = \frac{1}{2} CV^2$ $\frac{CV_0-q}{C} = \frac{q}{C/2} = \frac{2q}{C}$ $V_0 = \frac{3q}{C} \Rightarrow q = \frac{CV_0}{3}$ $U_i = \frac{1}{2} CV_0^2$ $U_f = \frac{\left(\frac{2CV_0}{3}\right)^2}{2C} + \frac{\left(\frac{CV_0}{3}\right)^2}{2(C/2)}$ $= \frac{1}{2} CV_0^2 \left[\frac{4}{9} + \frac{2}{9}\right] = \frac{1}{2} CV_0^2 \left(\frac{3}{3}\right)$ $Heat \ loss = \frac{1}{2} CV_0^2 - \left(\frac{2}{3}\right)\left(\frac{1}{2} CV_0^2\right)$ $= \frac{1}{6} CV_0^2$

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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
}