Check that the ratio $ke^{2} / G m_{e} m_{p}$ is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?

(N/A) The given ratio is $\frac{k e^{2}}{G m_{e} m_{p}}$.
Where,$G$ is the gravitational constant with unit $N \, m^{2} \, kg^{-2}$.
$m_{e}$ and $m_{p}$ are the masses of an electron and a proton,respectively,with unit $kg$.
$e$ is the electric charge with unit $C$.
$k = \frac{1}{4 \pi \varepsilon_{0}}$ is the Coulomb constant with unit $N \, m^{2} \, C^{-2}$.
Substituting the units:
$\frac{[N \, m^{2} \, C^{-2}] [C^{2}]}{[N \, m^{2} \, kg^{-2}] [kg] [kg]} = \frac{N \, m^{2}}{N \, m^{2}} = M^{0} L^{0} T^{0}$.
Thus,the ratio is dimensionless.
Using the values:
$k = 9 \times 10^{9} \, N \, m^{2} \, C^{-2}$
$e = 1.6 \times 10^{-19} \, C$
$G = 6.67 \times 10^{-11} \, N \, m^{2} \, kg^{-2}$
$m_{e} = 9.11 \times 10^{-31} \, kg$
$m_{p} = 1.67 \times 10^{-27} \, kg$
Calculating the value:
$\frac{9 \times 10^{9} \times (1.6 \times 10^{-19})^{2}}{6.67 \times 10^{-11} \times 9.11 \times 10^{-31} \times 1.67 \times 10^{-27}} \approx 2.3 \times 10^{39}$.
This ratio signifies the strength of the electrostatic force relative to the gravitational force between an electron and a proton.