The separation between an electron and a proton in a hydrogen atom is…

The correct answer is Option (3) → \( 33.39 \times 10^{-9}\,\mathrm{N} \)

The electron and the proton carry equal and opposite charges \( e \), so they attract each other with the force given by Coulomb's law.

    \( \displaystyle F = \frac{1}{4\pi\varepsilon_0}\,\frac{e^{2}}{r^{2}} \)

    \( r = 0.83\,\mathring{\mathrm{A}} = 0.83\times10^{-10}\,\mathrm{m} \)
    \( e = 1.6\times10^{-19}\,\mathrm{C},\ 1/4\pi\varepsilon_0 = 9\times10^{9}\,\mathrm{N\,m^2\,C^{-2}} \)

    \( \displaystyle F = \frac{9\times10^{9} \times (1.6\times10^{-19})^{2}}{(0.83\times10^{-10})^{2}} = \frac{2.304\times10^{-28}}{6.889\times10^{-21}} \)

    \( \displaystyle F \approx 3.34\times10^{-8}\,\mathrm{N} = 33.4\times10^{-9}\,\mathrm{N} \)

This matches \( 33.39\times10^{-9}\,\mathrm{N} \); the small difference comes from rounding the constants.