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Current Question (ID: 20132)

Question:

$\text{Two identical electric point dipoles have dipole moments } \vec{P}_1 = P\hat{i} \text{ and } \vec{P}_2 = -P\hat{i} \text{ and are held on the } x \text{ axis at distance } 'a' \text{ from each other.}$ $\text{When released, they move along the } x\text{-axis with the direction of their dipole moments remaining unchanged.}$ $\text{If the mass of each dipole is } 'm', \text{ their speed when they are infinitely far apart is:}$

Options:

1. $\frac{P}{a} \sqrt{\frac{1}{\pi \varepsilon_0 ma}}$
2. $\frac{P}{a} \sqrt{\frac{3}{2\pi \varepsilon_0 ma}}$
3. $\frac{P}{a} \sqrt{\frac{1}{2\pi \varepsilon_0 ma}}$
4. $\frac{P}{a} \sqrt{\frac{2}{\pi \varepsilon_0 ma}}$

Solution:

$\text{Hint: Apply the conservation of energy.}$ $\text{Using energy conservation}$ $KE_i + PE_i = KE_f + PE_f$ $\text{O} + \frac{2KP}{a^3} \times P = \frac{1}{2}mv^2 \times 2 + 0$ $V = \sqrt{\frac{2P^2}{4\pi \varepsilon_0 a^3 m}} = \frac{P}{a} \sqrt{\frac{1}{2\pi \varepsilon_0 am}}$

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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
}