Question

Check that the ratio `(ke^2)/(G m_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 of the electrostatic force to the gravitational force between an electron and a proton separated by a distance r is given by the following expression.

`(F_"elec")/(F_"grav") = (K_(e^2)/r^2)/((G m_e m_p)/r^2)`

= `(Ke^2)/(Gm_em_p)`

In terms of dimensions,

`[(Ke^2)/(Gm_em_p)] = [(F_"elec")/(F_"grav")]`

= `(MLT^-2)/(MLT^-2)`

= 1

Thus, the ratio `(Ke^2)/(Gm_em_p)` is dimensionless.

`(Ke^2)/(Gm_em_p) = (9 xx 10^9 xx (1.6 xx 10^-19)^2)/(6.67 xx 10^-11 xx 9.1 xx 10^-31 kg xx 1.67 xx 10^-27 kg)`

= `(9 xx 10^9 xx 2.56 xx 10^-38)/(101.36 xx 10^-69)`

= `(23.04 xx 10^-29)/(101.36 xx 10^-69)`

= 0.23 × 10⁴⁰

= 2.3 × 10³⁹

This ratio indicates that electrostatic forces are about 10³⁹ times stronger than gravitational forces.