Question

Check that the ratio \( ke^2/Gm_em_p \) is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?

Answer 1

The ratio to be determined is given as follows : \( \frac{ke^{2}}{Gm_{e}m_{p}} \)
where G is the gravitational constant in \( N m^2 kg^{-2} \)

\( m^e \) and \( m^p \) is the masses of electron and proton in kg.

e is the electric charge (unit – C)

\( k = \frac{1}{4\pi \epsilon _{o}} = 9×10^9 \ \ \ (unit – Nm^2C^{-2}) \)

Therefore, the unit of given ratio,

\( \frac{ke^{2}}{Gm_{e}m_{p}} = \frac{[Nm^{2}C^{-2}][C^{-2}]}{[Nm^{2}kg^{-2}][kg][kg]} = M^0L^0T^0 \) 

So, the given ratio is dimensionless.
Given,
e = 1.6 x 10-19 C
G = 6.67 x 10-11 N m2 kg-2
me = 9.1 x 10-31 kg
mp = 1.66 x 10-27 kg

Putting the above values in the given ratio, we get

\( \frac{ke^{2}}{Gm_{e}m_{p}} = = \frac{9 \times 10^{9} \times (1.6 \times 10^{-19})^{2}}{6.67 \times 10^{-11} \times 9.1 \times 10^{-31} \times 1.67 \times 10^{-27}} = 2.3*10^{39} \)

So, the above ratio is the ratio of the electric force to the gravitational force between a proton and an electron when the distance between them is constant.

Electric force to the gravitational force between a proton and an electron when the distance between them is constant.