Import Question JSON

Current Question (ID: 16465)

Question:

$\text{The electric field in a certain region is acting radially outward and is given by } E = Aa. \text{ A charge contained in a sphere of radius } a \text{ centered at the origin of the field will be given by:}$

Options:

1. $4\pi \varepsilon_0 Aa^2$
2. $\varepsilon_0 Aa^2$
3. $4\pi \varepsilon_0 Aa^3$
4. $\varepsilon_0 Aa^3$

Solution:

$\text{Hint: } \phi_{\text{net}} = \frac{q_{\text{en}}}{\varepsilon_0}$ $\text{Step: Find the field in a sphere of radius } a \text{ centered at the origin}$ $E = Ar$ $\text{According to the question, the electric field varies as } E = Ar$ $\text{Here } r \text{ is the radial distance.}$ $\text{At } r = a, E = Aa \quad \ldots (1)$ $\text{The net flux emitted from a spherical surface of radius } a \text{ is } \phi_{\text{net}} = \frac{q_{\text{en}}}{\varepsilon_0}$ $\Rightarrow (Aa) \times (4\pi a^2) = \frac{q}{\varepsilon_0} \quad [\text{Using equation (1)}]$ $\Rightarrow q = 4\pi \varepsilon_0 Aa^3$ $\text{Hence, option (3) is the correct answer.}$

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
}