A system consists of two protons and one electron. They are separated by a distance of \( 1\,\mathring{\mathrm{A}} \) from each other. If the potential energy at infinity is taken to be zero, the potential energy of the system is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( -14.4\,\mathrm{eV} \)
The electrostatic potential energy of a system of point charges is the sum over every pair, \( U = \sum k\,q_iq_j/r_{ij} \). Here there are three pairs, all at \( r = 1\,\mathring{\mathrm{A}} \): the two protons repel (\( +e\cdot+e \)), and each proton attracts the electron (\( +e\cdot-e \)); the zero of energy is at infinity.
\( \displaystyle U = \frac{ke^{2}}{r}\left(+1 - 1 - 1\right) = -\frac{ke^{2}}{r} \)
\( \displaystyle \frac{ke^{2}}{r} = \frac{9\times10^{9} \times (1.6\times10^{-19})^{2}}{10^{-10}} = 2.30\times10^{-18}\,\mathrm{J} = \frac{2.30\times10^{-18}}{1.6\times10^{-19}}\,\mathrm{eV} = 14.4\,\mathrm{eV} \)
\( \displaystyle U = -14.4\,\mathrm{eV} \)