Import Question JSON

Current Question (ID: 20129)

Question:

$\text{A parallel plate capacitor has plate of length } l, \text{ width } w \text{ and separation of plates is } d. \text{ It is connected to a battery of emf } V. \text{ A dielectric slab of the same thickness } d \text{ and of dielectric constant is being inserted between the plates of the capacitor. At what length of the slab inside plates, will be energy stored in the capacitor be two times the initial energy stored?}$

Options:

1. $\frac{l}{4}$
2. $\frac{l}{2}$
3. $\frac{l}{3}$
4. $\frac{2l}{3}$

Solution:

$\text{Hint: } C_i = \frac{\varepsilon_0 A}{d}$ $C_i = \frac{\varepsilon_0 lw}{d}$ $C_f = C_1 + C_2$ $C_f = \frac{d}{K \varepsilon_0 wx} + \frac{\varepsilon_0 A_2}{d}$ $C_f = \frac{K \varepsilon_0 wx}{d} + \frac{\varepsilon_0 w(l-x)}{d}$ $C_f = 2C_i \Rightarrow \frac{K \varepsilon_0 wx}{x} + \frac{\varepsilon_0 w(l-x)}{d} = \frac{2\varepsilon_0 lw}{d}$ $4x + l = \frac{l}{3} x = 2l$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

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
}