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Current Question (ID: 16591)

Question:

$\text{A capacitor is charged by a battery. The battery is removed and another identical uncharged capacitor is connected in parallel. The total electrostatic energy of the resulting system:}$

Options:

1. $\text{increases by a factor of 4.}$
2. $\text{decreases by a factor of 2.}$
3. $\text{remains the same.}$
4. $\text{increases by a factor of 2.}$

Solution:

$\text{The total potential energy stored in a system of capacitors} = \sum \frac{1}{2} CV^2$ $\text{Also, the potential drop remains the same in parallel across both capacitors.}$ $\text{Initially the stored energy, } U_1 = \frac{1}{2} CV^2$ $\text{Finally, the potential drop across each capacitor will be } \frac{V}{2}.$ $\text{So, finally the stored energy:}$ $U_2 = \frac{1}{2} C \left( \frac{V}{2} \right)^2 + \frac{1}{2} C \left( \frac{V}{2} \right)^2$ $= \frac{1}{2} (2C) \left[ \frac{V}{2} \right]^2$ $= \left( \frac{1}{4} CV^2 \right) = \frac{U_1}{2}$

Import JSON File

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
}