Question

Consider an electron and a proton separated by a distance of 1.0 nm.
A.) Calculate the magnitude of the gravitational force between them.
B.) Calculate the magnitude of the electric force between them.
C.) explain how the ratio of these gravitational and electric forces would change if the distance were increased to 1.0 m

Answer 1

1) The gravitational force between the proton and the electron is given by:
$F_G=G \frac{m_p m_e}{r^2}$
where
G is the gravitational constant
$m_p=1.67 \cdot 10^{-27}kg$ is the mass of the proton
$m_e = 9.1 \cdot 10^{-31}kg$ is the mass of the electron
$r=1.0 nm=1.0 \cdot 10^{-9} m$ is their separation

Substituting the numbers, we have
$F=(6.67 \cdot 10^{-11}) \frac{(1.67 \cdot 10^{-27})(9.1 \cdot 10^{-31})}{(1.0 \cdot 10^{-9})^2}=1.0 \cdot 10^{-49} N$

2) The electric force between the proton and the electron is given by:
$F_e=k \frac{q_p q_e}{r^2}=k \frac{e^2}{r^2} $
where
k is the Coulomb's constant
$q_p=q_e=e=1.6 \cdot 10^{-19}C$ is the charge of the proton and of the electron (it is the same)
$r=1.0 nm=1.0 \cdot 10^{-9} m$ is their separation

Substituting the numbers into the equation, we get
$F_e=(8.99 \cdot 10^9 ) \frac{(1.6 \cdot 10^{-19})^2}{(1.0 \cdot 10^{-9})^2}=2.3 \cdot 10^{-10}N$

C) The ratio between the two forces is therefore
$\frac{F_e}{F_g}= \frac{2.3 \cdot 10^{-10} N}{1.0 \cdot 10^{-49}N}=2.3 \cdot 10^{39}$
This ratio would not change if we change the distance between the two particles: in fact, both forces have the same dependance on the distance (as $1/r^2$), therefore they change by the same proportion, so their ratio remains the same.