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Current Question (ID: 10780)

Question:

$\text{If the radius of a star is } R \text{ and it acts as a black body, what would be the temperature of the star at which the rate of energy production is } Q?$ $(\sigma \text{ is Stefan-Boltzmann constant})$

Options:

1. $\frac{Q}{4\pi R^2 \sigma}$
2. $\left(\frac{Q}{4\pi R^2 \sigma}\right)^{\frac{1}{2}}$
3. $\left(\frac{4\pi R^2 Q}{\sigma}\right)^{\frac{1}{4}}$
4. $\left(\frac{Q}{4\pi R^2 \sigma}\right)^{\frac{1}{4}}$ (Correct)

Solution:

$\text{From Stefan's law:}$ $\text{For a black body, the rate of energy production (power radiated) is given by:}$ $Q = 4\pi R^2 \sigma T^4$ $\text{where:}$ $Q = \text{rate of energy production}$ $R = \text{radius of the star}$ $\sigma = \text{Stefan-Boltzmann constant}$ $T = \text{temperature of the star}$ $\text{So, the rate of energy production:}$ $Q = 4\pi R^2 \sigma T^4$ $\text{Solving for temperature } T:$ $T^4 = \frac{Q}{4\pi R^2 \sigma}$ $\text{The temperature of the star:}$ $T = \left(\frac{Q}{4\pi R^2 \sigma}\right)^{\frac{1}{4}}$

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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
}