Check that the ratio $k e^{2} / G m_{e} m_{p}$ is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?
The given ratio is $\frac{k e^{2}}{\mathrm{G} m_{\mathrm{e}} m_{\mathrm{p}}}$.
Where,
G = Gravitational constant
Its unit is $\mathrm{N} \mathrm{m}^{2} \mathrm{~kg}^{-2}$.\
$m_{\mathrm{e}}$ and $m_{\mathrm{p}}=$ Masses of electron and proton.
Their unit is kg.
$e=$ Electric charge.
Its unit is C.
$k=\mathrm{A}$ constant
$=\frac{1}{4 \pi \in_{0}}$
$\epsilon_{0}=$ Permittivity of free space
Its unit is $\mathrm{N} \mathrm{m}^{2} \mathrm{C}^{-2}$.
Therefore, unit of the given ratio $\frac{k e^{2}}{\mathrm{G} m_{\mathrm{e}} m_{\mathrm{p}}}=\frac{\left[\mathrm{Nm}^{2} \mathrm{C}^{-2}\right]\left[\mathrm{C}^{-2}\right]}{\left[\mathrm{Nm}^{2} \mathrm{~kg}^{-2}\right][\mathrm{kg}][\mathrm{kg}]}$
$=\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}$
Hence, the given ratio is dimensionless.
$e=1.6 \times 10^{-19} \mathrm{C}$
$G=6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2} \mathrm{~kg}^{-2}$
$m_{\mathrm{e}}=9.1 \times 10^{-31} \mathrm{~kg}$
$m_{\mathrm{p}}=1.66 \times 10^{-27} \mathrm{~kg}$
Hence, the numerical value of the given ratio is
$\frac{k e^{2}}{\mathrm{G} m_{e} m_{p}}=\frac{9 \times 10^{9} \times\left(1.6 \times 10^{-19}\right)^{2}}{6.67 \times 10^{-11} \times 9.1 \times 10^{-3} \times 1.67 \times 10^{-22}} \approx 2.3 \times 10^{39}$
This is the ratio of electric force to the gravitational force between a proton and an electron, keeping distance between them constant.
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