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Current Question (ID: 20175)

Question:

$\text{A parallel plate capacitor with plate area } A \text{ and plate separation } d = 2 \text{ m has a capacitance of } 4 \, \mu\text{F.}$ $\text{The new capacitance of the system if half of the space between them is filled with a dielectric material of dielectric constant } K = 3 \text{ (as shown in the figure) will be:}$

Options:

1. $2 \, \mu\text{F}$
2. $32 \, \mu\text{F}$
3. $6 \, \mu\text{F}$
4. $8 \, \mu\text{F}$

Solution:

$\text{The capacitance of a parallel plate capacitor is given by } C = \frac{K \varepsilon_0 A}{d}.$ $\text{When half of the space is filled with a dielectric, the system can be considered as two capacitors in parallel:}$ $C_1 = \frac{\varepsilon_0 A}{d/2} = \frac{2 \varepsilon_0 A}{d}, \quad C_2 = \frac{K \varepsilon_0 A}{d/2} = \frac{2K \varepsilon_0 A}{d}.$ $\text{The total capacitance is } C = C_1 + C_2 = \frac{2 \varepsilon_0 A}{d} + \frac{2K \varepsilon_0 A}{d} = \frac{2 \varepsilon_0 A}{d}(1 + K).$ $\text{Substituting } K = 3, \text{ we get } C = \frac{2 \varepsilon_0 A}{d}(1 + 3) = \frac{8 \varepsilon_0 A}{d}.$ $\text{Given } C_0 = \frac{\varepsilon_0 A}{d} = 4 \, \mu\text{F}, \text{ so } C = 8 \, \mu\text{F}.$

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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
}