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Current Question (ID: 20497)

Question:

$\text{An infinitely long straight wire carrying current } I, \text{ one side opened rectangular loop and a conductor } C \text{ with a sliding connector are located in the same plane, as shown in the figure. The connector has length } l \text{ and resistance } R. \text{ It slides to the right with a velocity } v. \text{ The resistance of the conductor and the self inductance of the loop are negligible. The induced current in the loop, as a function of separation } r, \text{ between the connector and the straight wire is:}$

Options:

1. $\frac{\mu_0}{\pi} \frac{Ivl}{Rr}$
2. $\frac{\mu_0}{2\pi} \frac{Ivl}{Rr}$
3. $2\pi \frac{Ivr}{Rr}$
4. $\frac{\pi}{\mu_0} \frac{Rr}{Ivl}$

Solution:

$\text{Hint: } i = \frac{\varepsilon_{\text{induced}}}{R}$ $B = \frac{\mu_0 i}{2\pi r}$ $\phi = \frac{\mu_0 i}{2\pi r} \cdot l \cdot dr$ $\Rightarrow \frac{d\phi}{dt} = \frac{\mu_0 i l}{2\pi r} \cdot \frac{dr}{dt}$ $e = \frac{\mu_0}{2\pi} \cdot \frac{Ivl}{Rr}$ $i = \frac{e}{R} = \frac{\mu_0}{2\pi} \cdot \frac{Ivl}{Rr}$

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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
}