Import Question JSON

Current Question (ID: 16973)

Question:

$\text{A uniform but time-varying magnetic field } B(t) \text{ exists in a circular region of radius } a \text{ and is directed into the plane of the paper, as shown. The magnitude of the induced electric field at point } P \text{ at a distance } r \text{ from the centre of the circular region:}$

Options:

1. $\text{is zero}$
2. $\text{decreases as } \frac{1}{r}$
3. $\text{increases as } r$
4. $\text{decreases as } \frac{1}{r^2}$

Solution:

$\text{Construct a concentric circle of radius } r. \text{ The induced electric field } (E) \text{ at any point on the circle is equal to that at } P. \text{ For this circle:}$ $\oint \vec{E} \cdot d\vec{l} = A \left| \frac{d\Phi}{dt} \right| = \pi a^2 \left| \frac{dB}{dt} \right|$ $\text{or } E \times (2\pi r) = \pi a^2 \left| \frac{dB}{dt} \right|$ $\Rightarrow E = \frac{a^2}{2r} \left| \frac{dB}{dt} \right| \Rightarrow E \propto \frac{1}{r}$

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
}