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Current Question (ID: 16877)

Question:

$\text{Figure shows two small identical magnetic dipoles } a \text{ and } b \text{ of magnetic moments } M \text{ each, placed at a separation } 2d, \text{ with their axes perpendicular to each other. The magnetic field at the point } P \text{ midway between the dipoles is:}$

Options:

1. $\frac{2\mu_0 M}{4\pi d^3}$
2. $\frac{\mu_0 M}{4\pi d^3}$
3. $\text{zero}$
4. $\frac{\sqrt{5} \mu_0 M}{4\pi d^3}$

Solution:

$\text{Hint: } B_{\text{net}} = \sqrt{B_1^2 + B_2^2}$ $\text{Step 1: Find the magnetic field due to individual dipoles.}$ $B_1 = 2\left(\frac{\mu_0}{4\pi}\right)\frac{M}{d^3}$ $B_2 = \left(\frac{\mu_0}{4\pi}\right)\frac{M}{d^3}$ $\text{Step 2: Find the net magnetic field.}$ $B_N = \sqrt{B_1^2 + B_2^2}$ $= \sqrt{5}\left[\frac{\mu_0 M}{4\pi d^3}\right]$

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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
}