Import Question JSON

Current Question (ID: 20221)

Question:

$\text{Model a torch battery of length } l \text{ to be made up of a thin cylindrical bar of radius } a \text{ and a concentric thin cylindrical shell of radius } b \text{ filled in between with an electrolyte of resistivity } \rho \text{ (see figure).}$ $\text{If the battery is connected to a resistance of value } R, \text{ the maximum Joule heating in } R \text{ will take place for:}$

Options:

1. $R = \frac{2\rho}{\pi l} \ln \left( \frac{b}{a} \right)$
2. $R = \frac{\rho}{\pi l} \ln \left( \frac{b}{a} \right)$
3. $R = \frac{\rho}{2\pi l} \left( \frac{b}{a} \right)$
4. $R = \frac{\rho}{2\pi l} \ln \left( \frac{b}{a} \right)$

Solution:

$\text{Hint: Maximum Power in the battery when, } r = R.$ $P_R = \left( \frac{\epsilon}{r + R} \right)^2 R$ $\rho_R \text{ is max. when } r = R$ $\int dr = \int_a^b \frac{\rho \, dr}{2\pi l} = r = \frac{\rho}{2\pi l} \ln \frac{b}{a}$

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
}