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Check that the ratio $k e^2 / G m_e m_p$ is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?
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An electron and a proton have charges of $1.6 \times 10^{-19} \mathrm{C}$ each and their masses are $9.1 \times 10^{-31} \mathrm{~kg}$ and $1.6 \times 10^{-27} \mathrm{~kg}$ respectively.
Universal Gravitational Constant
$$
\mathrm{G}=6.7 \times 10^{-11} \mathrm{Nm}^2 / \mathrm{kg}^2
$$
and $\mathrm{k}=\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2$
Electrostatic force between electron and proton,
$$
\mathrm{F}_{\mathrm{e}}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{e} . \mathrm{e}}{\mathrm{r}^2}
$$
where $r$ is the distance between their centres. Gravitational force between electron and proton,
$$
\mathrm{F}_{\mathrm{g}}=\frac{\mathrm{G} \cdot \mathrm{m}_{\mathrm{e}} \mathrm{m}_{\mathrm{p}}}{\mathrm{r}^2}
$$
Dividing we get,
$$
\frac{\mathrm{F}_{\mathrm{e}}}{\mathrm{F}_{\mathrm{g}}}=\frac{\mathrm{e}^2}{4 \pi \varepsilon_0 \mathrm{Gm}_{\mathrm{e}} \mathrm{m}_{\mathrm{p}}} \approx 2.4 \times 10^{39} \text {. }
$$
The ratio is quite huge. This shows that electrostatic forces are much stronger than gravitational forces. A common example is the lifting of a piece of a paper by charged comb against the force of entire earth on that paper.
Universal Gravitational Constant
$$
\mathrm{G}=6.7 \times 10^{-11} \mathrm{Nm}^2 / \mathrm{kg}^2
$$
and $\mathrm{k}=\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2$
Electrostatic force between electron and proton,
$$
\mathrm{F}_{\mathrm{e}}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{e} . \mathrm{e}}{\mathrm{r}^2}
$$
where $r$ is the distance between their centres. Gravitational force between electron and proton,
$$
\mathrm{F}_{\mathrm{g}}=\frac{\mathrm{G} \cdot \mathrm{m}_{\mathrm{e}} \mathrm{m}_{\mathrm{p}}}{\mathrm{r}^2}
$$
Dividing we get,
$$
\frac{\mathrm{F}_{\mathrm{e}}}{\mathrm{F}_{\mathrm{g}}}=\frac{\mathrm{e}^2}{4 \pi \varepsilon_0 \mathrm{Gm}_{\mathrm{e}} \mathrm{m}_{\mathrm{p}}} \approx 2.4 \times 10^{39} \text {. }
$$
The ratio is quite huge. This shows that electrostatic forces are much stronger than gravitational forces. A common example is the lifting of a piece of a paper by charged comb against the force of entire earth on that paper.
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