Question

The masses of the earth and moon are 5.98 x 1024 and 7.35 x 1022 kg, respectively. Identical amounts of charge are placed on each body, such that the net force (gravitational plus electrical) on each is zero. What is the magnitude of the charge placed on each body?

Answer 1

Answer:

$Q\ =\ 5.70\times 10^{13}\ C.$

Explanation:

Given,

• Mass of the earth = $m_1\ =\ 5.98\times 10^{24}\ kg$
• Mass of the moon = $m_2\ =\ 7.35\times 10^{22}\ kg$
• universal gravitational constant = G = $6.67\times 10^{-11}\ m^2kg^{-1}s^{-2}$

Let Q be charges on the both earth and the moon, and 'r' be the distance between the earth and the moon.

Gravitational force between the earth and the moon is  attractive while electrical force between the earth and the moon is repulsive due to identical charges. Hence both are opposite in the sing but are equal in magnitude

$\therefore \dfrac{Gm_1m_2}{r^2}\ +\ \dfrac{-kQ^2}{r^2}\\\Rightarrow Gm_1m_2\ =\ kQ^2\\\Rightarrow Q\ =\ \sqrt{\dfrac{Gm_1m_2}{k}}\\\Rightarrow Q\ =\ \sqrt{\dfrac{6.67\times 10^{-11}\times 5.98\times 10^{24}\times 7.35\times 10^{22}}{9\times 10^9}}\\\Rightarow 5.707\times 10^{13}\ C$

Hence, the charges on both the earth and the moon are the same as of $5.707\times 10^{13}\ C.$