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Current Question (ID: 16887)

Question:

$\text{Two short magnets of equal dipole moments } M \text{ are fastened perpendicularly at their centres (figure). The magnitude of the magnetic field at a distance } d \text{ from the centre on the bisector of the right angle is:}$

Options:

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

Solution:

$\text{Hint: } M_{\text{net}} = \sqrt{M^2 + M^2} = \sqrt{2}M$ $\text{Step: Find the magnitude of the magnetic field at a distance } d \text{ from the centre on the bisector of the right angle.}$ $\text{The resultant magnetic moment of the two magnets is given by:}$ $M_{\text{net}} = \sqrt{M^2 + M^2} = \sqrt{2}M$ $\text{Imagine a short magnet lying along with } OP \text{ with a magnetic moment equal to } M\sqrt{2}. \text{ Thus, the point } P \text{ lies on the axial line of the magnet.}$ $\text{The magnitude of the magnetic field at the axial point is given by:}$ $\Rightarrow B = \frac{\mu_0}{4\pi} \frac{2M}{d^3}$ $\text{So, the magnitude of the magnetic field at } P \text{ is given by:}$ $\Rightarrow B = \frac{\mu_0}{4\pi} \frac{2\sqrt{2}M}{d^3}$ $\text{Hence, option (3) is the correct answer.}$

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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
}