The masses of the earth and moon are 5.98 × 1024 and 7.35 × 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?
To find the magnitude of the charge placed on each body, we need to equate the gravitational force with the electrical force.
First, we can find the gravitational force between the Earth and the Moon using Newton's Law of Universal Gravitation:
F_gravity = (G * m1 * m2) / r^2
Where:
F_gravity = gravitational force between the Earth and the Moon
G = universal gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1 = mass of the Earth (5.98 × 10^24 kg)
m2 = mass of the Moon (7.35 × 10^22 kg)
r = distance between the Earth and the Moon (approximately 384,400,000 m)
Substituting the given values into the equation, we can calculate the gravitational force between the Earth and the Moon.
F_gravity = (6.67430 × 10^-11 * 5.98 × 10^24 * 7.35 × 10^22) / (384,400,000)^2
Next, we can consider the electrical force, which is given as zero. If the net electrostatic force on each body is zero, it means that the electrical force is equal in magnitude but opposite in direction to the gravitational force:
F_electric = -F_gravity
To solve for the magnitude of the electrical force, we use Coulomb's Law:
F_electric = (k * |q1 * q2|) / r^2
Where:
F_electric = electrical force
k = Coulomb's constant (approximated as 9 × 10^9 N m^2/C^2)
q1 = charge on the Earth
q2 = charge on the Moon
r = distance between the Earth and the Moon (384,400,000 m)
Substituting the values of F_gravity and F_electric into the equation, we have:
(9 × 10^9 * |q1 * q2|) / (384,400,000)^2 = (6.67430 × 10^-11 * 5.98 × 10^24 * 7.35 × 10^22) / (384,400,000)^2
We can cancel out the common terms on both sides of the equation:
9 × 10^9 * |q1 * q2| = 6.67430 × 10^-11 * 5.98 × 10^24 * 7.35 × 10^22
Now, we can solve for the magnitude of the charge on each body, which is |q1| and |q2|:
|q1 * q2| = (6.67430 × 10^-11 * 5.98 × 10^24 * 7.35 × 10^22) / (9 × 10^9)
Taking the square root of both sides:
|q1| = √[(6.67430 × 10^-11 * 5.98 × 10^24 * 7.35 × 10^22) / (9 × 10^9)]
Calculating this value will yield the magnitude of the charge placed on each body.