Question

Check that the ratio $\frac{ke^2}{ Gm_e m_p}$ is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?

Answer 1

The given ratio is $\frac{ke^2}{ Gm_e m_p}$. Where,
G = Gravitational constant. Its unit is $Nm^2kg^{−2}$
$m_e$ and $m_p$ = Masses of electron and proton and their unit is kg.
e = Electric charge. Its unit is C.
$k = \frac{1}{4πε_0}$ and its unit is $Nm^2C^{−2}$
Therefore, unit of the given ratio
$\frac{ke^2}{ Gm_e m_p}$ = $\frac{[Nm^2C^{−2}][C^{-2}]}{[Nm^2kg^{−2}][Kg][Kg]} = M^0L^0 T^0$
Hence, the given ratio is dimensionless.
e = $1.6 × 10^{−19}C$
G = $6.67 × 10−11Nm^2kg^{−2}$
$m_e=$ $9.1 × 10^{−31} kg$
$m_p = 1.66 × 10^{−27} kg$
Hence, the numerical value of the given ratio is
$\frac{ke^2}{ Gm_e m_p}$ $= \frac{9 × 10^9 × (1.6 × 10^{-19})^{2}} {6.67 × 10^{-11} × 9.1 × 10^{ -31}× 1.67 × 10^{ -27 }}≈ 2.3 × 10^{39}$
This is the ratio of electric force to the gravitational force between a proton and an electron, keeping distance between them constant.