Check that the ratio ke²/G mₑmₚ 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 ke² /G m_em_p

Where, G = Gravitational constant. Its unit is Nm² kg^-2

m_e and m_p = Masses of electron and proton and their unit is kg.

e=Electric charge and its unit is C.

k =1 /4πε₀and its unit  Nm²C^-2.

Therefore, unit of the given ratio

ke²/Gm_em_p= [Nm²C^-2][ C^-2]/[Nm²kg^-2][ kg] [ kg]

=M⁰L⁰T⁰

Hence, the given ratio is dimensionless.

e = 1.6 x 10^-19 C

G = 6.67 x 10^-11 N m² kg’²

m_e= 9.1 x 10-31 kg

m_p = 1.66 x 10^-27kg

Hence, the numerical value of the given ratio is

ke² / G m_em_p=      9 x 10⁹ x (1.6 x 10^-19)²/ 6.67×10^-11 x 9.1×10^-31x 1.67×10^-27

≈2.3 x 10³⁹

This is the ratio of electric force to the gravitational force between a proton and an electron, keeping distance between them constant.