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Current Question (ID: 16549)

Question:

$\text{Three capacitors of capacitances } 3 \, \mu\text{F}, 9 \, \mu\text{F} \text{ and } 18 \, \mu\text{F} \text{ are connected once in series and another time in parallel.}$ $\text{The ratio of equivalent capacitance in the two cases } \frac{C_s}{C_p} \text{ will be:}$

Options:

1. $1 : 15$
2. $15 : 1$
3. $1 : 1$
4. $1 : 3$

Solution:

$\text{Hint: } \frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$ $\text{Step: Find the ratio of equivalent capacitance in the two cases } \frac{C_s}{C_p}$ $\text{For capacitors in series, the formula for equivalent capacitance } C_s \text{ is given by:}$ $\frac{1}{C_s} = \frac{1}{3} + \frac{1}{9} + \frac{1}{18}$ $\Rightarrow C_s = \frac{18}{9} = 2 \, \mu\text{F}$ $\text{For capacitors in parallel, the formula for equivalent capacitance } C_p \text{ is given by:}$ $C_p = C_1 + C_2 + C_3$ $C_p = 3 \, \mu\text{F} + 9 \, \mu\text{F} + 18 \, \mu\text{F}$ $\Rightarrow C_p = 3 + 9 + 18 = 30 \, \mu\text{F}$ $\text{Now, we need to find the ratio } \frac{C_s}{C_p}:$ $\Rightarrow \frac{C_s}{C_p} = \frac{2 \, \mu\text{F}}{30 \, \mu\text{F}} = \frac{2}{30} = \frac{1}{15}$ $\text{Hence, option (1) is the correct answer.}$

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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
}