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Current Question (ID: 20138)

Question:

$\text{A parallel plate capacitor is formed by two plates, each of area } 30\pi \ \text{cm}^2 \text{ separated by 1 mm. A material of dielectric strength } 3.6 \times 10^7 \ \text{Vm}^{-1} \text{ is filled between the plates. If the maximum charge that can be stored on the capacitor without causing any dielectric breakdown is } 7 \times 10^{-6} \ \text{C, the value of the dielectric constant of the material is:}$ $\left\{ \text{Use: } \frac{1}{4\pi \varepsilon_0} = 9 \times 10^9 \ \text{Nm}^2\text{C}^{-2} \right\}$

Options:

1. $1.66$
2. $1.75$
3. $2.25$
4. $2.33$

Solution:

$\text{Hint: } Q = CV$ $\text{Step 1: Find the surface charge density.}$ $\sigma = \frac{Q_{\text{max}}}{A} = \frac{7 \times 10^{-6}}{30\pi \times 10^{-4}} = \frac{7 \times 10^{-3}}{3\pi}$ $\text{Step 2: Find the value of the dielectric constant of the material.}$ $E_{\text{max}} = \frac{\sigma}{K\varepsilon_0}$ $\Rightarrow K = \frac{\sigma}{E_{\text{max}}\varepsilon_0}$ $\Rightarrow K = \frac{Q_{\text{max}}}{AE_{\text{max}}\varepsilon_0} = \frac{7 \times 10^{-3}}{3\pi \times 3.6 \times 10^7 \times \varepsilon_0} = 2.33$

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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
}