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Current Question (ID: 20145)

Question:

$\text{A source of potential difference } V \text{ is connected to the combination of two identical capacitors as shown in the figure.}$ $\text{When key } 'K' \text{ is closed, the total energy stored across the combination is } E_1.$ $\text{Now key } 'K' \text{ is opened and dielectric of dielectric constant } 5 \text{ is introduced between the plates of the capacitors.}$ $\text{The total energy stored across the combination is now } E_2.$ $\text{The ratio } \frac{E_1}{E_2} \text{ will be:}$

Options:

1. $\frac{1}{10}$
2. $\frac{2}{5}$
3. $\frac{5}{13}$
4. $\frac{5}{26}$

Solution:

$\text{Hint: } E = \frac{1}{2}CV^2$ $\text{Step 1: Find the initial energy stored in the combination.}$ $E_1 = \frac{1}{2}CV^2 \times 2 = CV^2$ $\text{Step 2: Find the final energy stored in the combination.}$ $\text{When switch is opened charge on right capacitor remain constant,}$ $\text{while potential on left capacitor remain same.}$ $E_2 = k\left(\frac{CV^2}{2}\right) + \frac{CV^2}{2k}$ $= \frac{1}{2}CV^2\left(5 + \frac{1}{5}\right)$ $= \frac{26}{10}CV^2$ $\text{Step 3: Find the ratio of energy stored.}$ $\frac{E_1}{E_2} = CV^2 \times \frac{10}{26CV^2}$ $= \frac{5}{13}$

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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
}