Import Question JSON

Current Question (ID: 20414)

Question:

$\text{The magnetic field at the centre of a circular coil of radius } r, \text{ due to current } I \text{ flowing through it, is } B. \text{ The magnetic field at a point along the axis at a distance } \frac{r}{2} \text{ from the centre is:}$

Options:

1. $\frac{B}{2}$
2. $2B$
3. $\left(\frac{2}{\sqrt{5}}\right)^3 B$
4. $\left(\frac{2}{\sqrt{3}}\right)^3 B$

Solution:

$\text{Using the formula for the magnetic field at a point on the axis of a circular coil:}$ $B_{\text{axis}} = \frac{B}{2} \left(\frac{r^2}{r^2 + x^2}\right)^{3/2}$ $\text{where } x = \frac{r}{2}$ $\Rightarrow B_{\text{axis}} = \frac{B}{2} \left(\frac{r^2}{r^2 + \left(\frac{r}{2}\right)^2}\right)^{3/2}$ $= \frac{B}{2} \left(\frac{r^2}{r^2 + \frac{r^2}{4}}\right)^{3/2}$ $= \frac{B}{2} \left(\frac{r^2}{\frac{5r^2}{4}}\right)^{3/2}$ $= \frac{B}{2} \left(\frac{4}{5}\right)^{3/2}$ $= \left(\frac{2}{\sqrt{5}}\right)^3 B$

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
}