how many electrons must be removed from each of the two 5kg copper spheres to make the electric force of repulsion between them equal in magnitude to the gravitational attraction between them?
To determine the number of electrons that must be removed from each copper sphere, we need to compare the electric force of repulsion between them to the gravitational attraction between them. Let's break down the steps to find the answer:
1. Calculate the gravitational force between the copper spheres:
We can use Newton's law of universal gravitation to find the gravitational force between two objects:
F_gravity = (G * m1 * m2) / r^2
where:
F_gravity is the gravitational force,
G is the gravitational constant (6.67430 × 10^-11 m^3/(kg s^2)),
m1 and m2 are the masses of the copper spheres (5 kg each),
r is the distance between the centers of the spheres.
2. Calculate the electric force of repulsion between the copper spheres:
The electric force of repulsion between two charged spheres can be found using Coulomb's law:
F_electric = k * (q1 * q2) / r^2
where:
F_electric is the electric force of repulsion,
k is the Coulomb constant (9 × 10^9 N m^2/C^2),
q1 and q2 are the charges on the copper spheres (to be determined),
r is the distance between the centers of the spheres.
3. Set the electric force equal to the gravitational force and solve for q1 and q2:
F_electric = F_gravity
k * (q1 * q2) / r^2 = (G * m1 * m2) / r^2
q1 * q2 = ((G * m1 * m2) / k)
4. Determine the number of electrons:
The charge (q) of an electron is -1.6 × 10^-19 C. Considering that each copper atom donates two electrons, the total charge (Q) can be calculated by multiplying the number of electrons (n) by the charge of each electron:
Q = n * (-1.6 × 10^-19 C)
Substituting Q and q into the equation, we get:
(n1 * (-1.6 × 10^-19 C)) * (n2 * (-1.6 × 10^-19 C)) = ((G * m1 * m2) / k)
Solving this equation will give us the number of electrons that must be removed from each copper sphere.
Note: The actual number of electrons may not be a whole number, as the result obtained is an approximation.