Check that the ratio `(k e^(2))/( Gm_em_p) ` is dimensionless. Look up a Table of Physical Constant and determine the value of this ratio. What does the ratio signify?

Explanation

From the relation `F= ( kq_1q_2)/(r^(2)) ` we find units of k as `Nm^(2) C^(2) `.
Again units of F are `Nm^(2) kg^(-2) `
` therefore `Units of ` (ke^(2))/( Gm_rm_p) = ( Nm^(2) C^(-2) (C^(2)))/( (Nm^(2) kg^(-2) )(kg)(kg)) = `units less and hence dimensionless.
We know that ` k= 9 xx 10 ^(9) Nm ^(2) C^(2) .,e= 1.6 xx 10 ^(-19) C , G = 6.67 xx 10 ^(-11) Nm ^(2) kg ^(-2), m _e = 9.1 xx 10 ^(-31) kg and m_p = 1.67 xx 10 ^(-27) kg `
` therefore ` value of `(k e^(2))/( Gm_e,m_p) =( 9xx 10^(9) xx (1.6 xx 10 ^(-19))^(2) )/( 6.67 xx 10 ^(-11) xx 9.1 xx10 ^(-31) xx 1. 67 xx 10 ^(-27)) = 2.4 xx 10 ^(39 ) `
The ratio signifies the ratio of electric force and the gravitational force acting between an electron and a proton separated by a finite distance.