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Current Question (ID: 16502)

Question:

$\text{A charge } +q \text{ is fixed at each of the points } x = x_0, x = 3x_0, x = 5x_0 \ldots \text{ infinite, on the x-axis, and a charge } -q \text{ is fixed at each of the points } x = 2x_0, x = 4x_0, x = 6x_0, \ldots \text{ infinite. Here } x_0 \text{ is a positive constant. Take the electric potential at a point due to a charge } Q \text{ at a distance } r \text{ from it to be } \frac{Q}{4\pi\varepsilon_0 r}. \text{ Then, the potential at the origin due to the above system of charges is:}$

Options:

1. $0$
2. $\frac{q}{8\pi\varepsilon_0 x_0 \ln 2}$
3. $\infty$
4. $\frac{q \ln 2}{4\pi\varepsilon_0 x_0}$

Solution:

$V = \frac{q}{4\pi\varepsilon_0 x_0} \left[ 1 + \frac{1}{3} + \frac{1}{5} + \ldots \right] - \frac{q}{4\pi\varepsilon_0 x_0} \left[ \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \ldots \right]$ $V = \frac{q}{4\pi\varepsilon_0 x_0} \left[ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots \right]$ $\text{Expansion of } \ln (1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} \ldots$ $\text{So, if } x = 1,$ $\ln 2 = \left[ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots \right]$ $V = \frac{q}{4\pi\varepsilon_0 x_0} \ln 2$

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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
}