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Current Question (ID: 20064)

Question:

$\text{An electron is moving with the speed } 1 \text{ m/s at a distance } 1 \text{ m from a large sheet of charge with density } \sigma \text{ C/m}^2. \text{ Then the maximum value of } \sigma \text{ such that the electron hit the sheet after } 1 \text{ sec is:}$ $\text{(mass of electron } 9 \times 10^{-31} \text{ kg, permittivity of free space } \varepsilon_0 = 9 \times 10^{-12} \text{ C}^2/\text{Nm}^2)$

Options:

1. $4.05 \times 10^{-22} \text{ C/m}^2$
2. $8.10 \times 10^{-22} \text{ C/m}^2$
3. $4.05 \times 10^{24} \text{ C/m}^2$
4. $2.02 \times 10^{-20} \text{ C/m}^2$

Solution:

$\text{Hint: } a = \frac{qE}{m}$ $\text{For maximum value of } \sigma, \text{ initially, electron must move away from plate.}$ $ut + \frac{1}{2}at^2 = s$ $t = 1 \quad u = 1 \text{ m/s} \quad s = -1 \text{ m}$ $1 \times 1 - \frac{1}{2}a \times 1^2 = -1$ $\Rightarrow \quad a = 4 \text{ m/s}^2$ $\frac{qE}{m} = 4$ $\frac{q\sigma}{2\varepsilon_0 m} = 4$ $\sigma = \frac{4 \times 2 \times 9 \times 10^{-12} \times 9 \times 10^{-31}}{1.6 \times 10^{-19}}$ $= \frac{8 \times 81}{1.6} \times 10^{-24}$ $= 4.05 \times 10^{-22} \text{C/m}^2$

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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
}