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Current Question (ID: 20241)

Question:

$\text{Two identical cells, each having EMF } E, \text{ are connected in series to an external resistance } R. \text{ The internal resistances of the cells are } r_1 \text{ and } r_2, \text{ with } r_1 > r_2. \text{ If the potential difference across the cell of internal resistance } r_1 \text{ is zero, then the value of } R \text{ is:}$

Options:

1. $(r_1 - r_2)$
2. $\frac{r_1 r_2}{(r_1 + r_2)}$
3. $\frac{(r_1 + r_2)}{2}$
4. $(r_2 - r_1)$

Solution:

$\text{Hint: } I = \frac{E}{R_{eq}}$ $\text{Step 1: Draw the circuit diagram.}$ $\text{Step 2: Find the EMF of the cell.}$ $V = E - Ir_1$ $\Rightarrow 0 = E - Ir_1$ $\Rightarrow E = Ir_1$ $\text{Step 3: Find the value of external resistance } R.$ $\text{Apply KVL to the given circuit:}$ $-IR - Ir_2 + \frac{E}{2} - Ir_1 + E = 0$ $\Rightarrow I = \frac{2 \times Ir_1}{R + r_1 + r_2}$ $\Rightarrow I = \frac{2 \times Ir_1}{R + r_1 + r_2} \quad [\because E = Ir_1]$ $\Rightarrow R = r_1 - r_2$ $\text{Hence, option (1) is the correct answer.}$

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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
}