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Current Question (ID: 20139)

Question:

$\text{If the charge on a capacitor is increased by } 2 \ C, \text{ the energy stored in it increases by } 44\%. \text{ The original charge on the capacitor (in Coulomb) is:}$ $1. \ 10$ $2. \ 20$ $3. \ 30$ $4. \ 40$

Options:

1. $10$
2. $20$
3. $30$
4. $40$

Solution:

$\text{Hint: } U = \frac{Q^2}{2C}$ $\text{Step: Find the original charge on the capacitor.}$ $\text{Let } Q_0 \text{ be the original charge on the capacitor. After increasing the charge by } 2 \ C, \text{ the new charge becomes } Q_1 = Q_0 + 2.$ $\text{The initial energy stored in the capacitor is given by:}$ $U_0 = \frac{1}{2} \frac{Q_0^2}{C}$ $\text{The new energy stored after increasing the charge is given by:}$ $U_1 = \frac{1}{2} \frac{(Q_0 + 2)^2}{C}$ $\text{According to the problem, the new energy increases by } 44\%. \text{ Therefore, we can express this relationship as:}$ $U_1 = U_0 + 0.44U_0 = 1.44U_0$ $\text{Substituting the expressions for } U_0 \text{ and } U_1:$ $\frac{1}{2} \frac{(Q_0 + 2)^2}{C} = 1.44 \left( \frac{1}{2} \frac{Q_0^2}{C} \right)$ $Q_0 + 2 = 1.2Q_0$ $\Rightarrow Q_0 = \frac{2}{0.2} = 10 \ C$ $\text{Hence, option (1) is the correct answer.}$

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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
}