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Current Question (ID: 20495)

Question:

$\text{A series } L-R \text{ circuit is connected to a battery of emf } V. \text{ If the circuit is switched on at } t = 0, \text{ then the time at which the energy stored in the inductor reaches } \left(\frac{1}{n}\right) \text{ times of its maximum value, is:}$

Options:

1. $\frac{L}{R} \ln \left(\frac{\sqrt{n-1}}{\sqrt{n}}\right)$
2. $\frac{L}{R} \ln \left(\frac{\sqrt{n}}{\sqrt{n-1}}\right)$
3. $\frac{L}{R} \ln \left(\frac{\sqrt{n}}{\sqrt{n+1}}\right)$
4. $\frac{L}{R} \ln \left(\frac{\sqrt{n+1}}{\sqrt{n-1}}\right)$

Solution:

$\text{Hint: } U_{\text{max}} = \frac{1}{2} L I_{\text{max}}^2$ $U_{\text{max}} = \frac{1}{2} L I_{\text{max}}^2$ $i = I_{\text{max}} \left(1 - e^{-Rt/L}\right)$ $\text{For } U \text{ to be } \frac{U_{\text{max}}}{n}, \; i \text{ has to be } \frac{I_{\text{max}}}{\sqrt{n}}$ $\frac{I_{\text{max}}}{\sqrt{n}} = I_{\text{max}} \left(1 - e^{-Rt/L}\right)$ $e^{-Rt/L} = 1 - \frac{1}{\sqrt{n}} = \frac{\sqrt{n} - 1}{\sqrt{n}}$ $-\frac{Rt}{L} = \ln \left(\frac{\sqrt{n} - 1}{\sqrt{n}}\right)$ $t = \frac{L}{R} \ln \left(\frac{\sqrt{n}}{\sqrt{n-1}}\right)$

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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
}