Import Question JSON

Current Question (ID: 10749)

Question:

$\text{Two rods P and Q of equal length and having cross-sections } A_P \text{ and } A_Q \text{ respectively, have the same temperature difference across their ends. If } k_P \text{ and } k_Q \text{ are their thermal conductivities, then the condition for their equal rate of conduction of heat will be:}$

Options:

1. $k_P A_P = k_Q A_Q$ (Correct)
2. $\frac{\sqrt{k_P}}{A_P} = \frac{\sqrt{k_Q}}{A_Q}$
3. $k_P A_Q = k_Q A_P$
4. $k_P^2 A_P = k_Q^2 A_Q$

Solution:

\text{Hint: Apply Newton's law of cooling.} \text{Step 1: The rate of cooling is given as:} \frac{dQ}{dt} = -kA\frac{\Delta T}{\Delta x} \text{Step 2: Find the required condition by applying Newton's law of cooling.} \text{For rod P: } \frac{dQ}{dt} = k_P A_P \frac{\Delta T}{\Delta x} \text{For rod Q: } \frac{dQ}{dt} = k_Q A_Q \frac{\Delta T}{\Delta x} \text{Since both rods have equal length, equal temperature difference, and equal rate of heat conduction:} k_P A_P \frac{\Delta T}{\Delta x} = k_Q A_Q \frac{\Delta T}{\Delta x} \text{Therefore: } k_P A_P = k_Q A_Q

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
}