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Question
Check that the ratio ke2/G memp is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?
Solution
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.
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