Import Question JSON

Current Question (ID: 10755)

Question:

$\text{Four rods of the same material with different radii } r \text{ and length } l \text{ are used to connect two heat reservoirs at different temperatures. In which of the following cases is the heat conduction fastest?}$

Options:

1. $r = \frac{1}{3} \text{ cm}, l = \frac{1}{9} \text{ cm}$
2. $r = 3 \text{ cm}, l = 9 \text{ cm}$
3. $r = 4 \text{ cm}, l = 8 \text{ cm}$ (Correct)
4. $r = 1 \text{ cm}, l = 1 \text{ cm}$

Solution:

$\text{Hint: Heat conduction is depending on the thermal resistance of the rod.}$ $\text{Step 1: Find the thermal resistance.}$ $\text{Heat current is given by,}$ $\frac{\Delta Q}{\Delta T} = \frac{\Delta T}{R_T}$ $R_T = \frac{l}{kA}$ $\text{Step 2: Check each option one by one.}$ $\text{1. } \frac{l}{r^2} = 1$ $\text{3. } \frac{l}{r^2} = \frac{1}{2}$ $\text{2. } \frac{l}{r^2} = 1$ $\text{4. } \frac{l}{r^2} = 1$ $\text{So, the option (3) is correct}$

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
}