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Current Question (ID: 16412)

Question:

$\text{The ratio of Coulomb's electrostatic force to the gravitational force between an}$ $\text{electron and a proton separated by some distance is } 2.4 \times 10^{39}. \text{ The ratio of}$ $\text{the proportionality constant, } k = \frac{1}{4\pi\varepsilon_0} \text{ to the gravitational constant } G \text{ is}$ $\text{nearly:}$ $\text{(Given that the charge of the proton and electron each } = 1.6 \times 10^{-19}, \text{ the}$ $\text{mass of the electron } = 9.11 \times 10^{-31} \text{ kg, the mass of the proton } = 1.67 \times 10^{-27} \text{ kg):}$

Options:

1. $10^{20}$
2. $10^{30}$
3. $10^{40}$
4. $10$

Solution:

$\text{Hint: } \frac{F_e}{F_G} = 2.4 \times 10^{39}$ $\text{Step: Find the ratio of the proportionality constant to the gravitational}$ $\text{constant.}$ $k = 9 \times 10^9, G = 6.67 \times 10^{-11}$ $\frac{F_e}{F_G} = 2.04 \times 10^{39}$ $\frac{ke^2}{Gm_e m_p} = 2.04 \times 10^{39}$ $\frac{k}{G} = \frac{2.4 \times 10^{39} \times (9.11 \times 10^{-31}) \times (1.67 \times 10^{-27})}{(1.6 \times 10^{-19})^2}$ $\Rightarrow \frac{k}{G} = 1.42 \times 10^{20} \approx 10^{20}$ $\text{Hence, option (1) is the correct answer.}$

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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
}