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Current Question (ID: 19962)

Question:

$\text{Two identical positive charges } Q \text{ each are fixed at a distance of } '2a' \text{ apart from each other. Another point charge } q_0 \text{ with mass } 'm' \text{ is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge } q_0 \text{ executes SHM. The time period of oscillation of charge } q_0 \text{ will be:}$

Options:

1. $\sqrt{\frac{4\pi^3 \varepsilon_0 m a^3}{q_0 Q}}$
2. $\sqrt{\frac{q_0 Q}{4\pi^3 \varepsilon_0 m a^3}}$
3. $\sqrt{\frac{2\pi^2 \varepsilon_0 m a^3}{q_0 Q}}$
4. $\sqrt{\frac{8\pi^3 \varepsilon_0 m a^3}{q_0 Q}}$

Solution:

$\text{The force on } q_0 \text{ due to each charge } Q \text{ is } F = \frac{1}{4\pi \varepsilon_0} \frac{q_0 Q}{(a)^2}$. $\text{Net force } F_{\text{net}} = 2F \cos(\theta) = \frac{2q_0 Q}{4\pi \varepsilon_0 a^2}$. $\text{For small displacement, } F_{\text{net}} \approx -kx$. $\text{Comparing with } F = ma$, $\text{we get } k = \frac{2q_0 Q}{4\pi \varepsilon_0 a^3}$. $\text{Time period } T = 2\pi \sqrt{\frac{m}{k}} = \sqrt{\frac{4\pi^3 \varepsilon_0 m a^3}{q_0 Q}}$

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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
}