Question

Check that the ratio ke²/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 is

`(F_"elec")/(F_"grav") = (K_(e.e)/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")`

= 0.23 × 10⁴⁰

= 2.3 × 10³⁹

That electrostatic forces are a factor of 10³⁹ more than gravitational forces is indicated by this ratio.