#### Question

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?

#### Solution

The given ratio is

`(ke^2)/(Gm_cm_p)`

Where,

G = Gravitational constant

Its unit is N m^{2 }kg^{−2}.

*m*_{e} and *m*_{p} = Masses of electron and proton.

Their unit is kg.

*e* = Electric charge.

Its unit is C.

k=A constant

`=1/(4piin_0)`

∈_{0} = Permittivity of free space

Its unit is N m^{2 }C^{−2}.

therefore, unit of the given ratio `(ke^2)/(Gm_cm_p)=([Nm^2C^-2][c^-2])/([Nm^2kg^-2][kg][kg])=m^0L^0T^0`

Hence, the given ratio is dimensionless.

*e *= 1.6 × 10^{−19 }C

G = 6.67 × 10^{−11} N m^{2 }kg^{-2}

*m*_{e}= 9.1 × 10^{−31 }kg

*m*_{p} = 1.66 × 10^{−27 }kg

Hence, the numerical value of the given ratio is

`(ke^2)/(Gm_cm_p)=(9xx10^9xx(1.6xx10^-19)^2)/(6.67xx10^-1xx9.1xx10^-3xx1.67xx10^-22)~~2.3xx10^39`

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