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Current Question (ID: 16682)

Question:

$\text{A current of } 2 \text{ A flows through a } 2 \ \Omega \text{ resistor when connected across a battery.}$ $\text{The same battery supplies a current of } 0.5 \text{ A when connected across a } 9 \ \Omega \text{ resistor.}$ $\text{The internal resistance of the battery is:}$

Options:

1. $\frac{1}{3} \ \Omega$
2. $\frac{1}{4} \ \Omega$
3. $1 \ \Omega$
4. $0.5 \ \Omega$

Solution:

$\text{(1) Hint: Apply Kirchoff's voltage law.}$ $\text{Step 1: Find the initial condition.}$ $\text{Let the internal resistance of the battery } = r$ $\text{Initially, } i_1 = 2 = \frac{E}{2+r} \quad \text{(i)}$ $\text{Step 2: Find the final condition.}$ $\text{Finally, } i_2 = 0.5 = \frac{E}{9+r} \quad \text{(ii)}$ $\text{Step 3: Solve both the equations.}$ $\frac{2}{0.5} = \frac{9+r}{2+r}$ $\Rightarrow r = \frac{1}{3} \ \Omega$

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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
}