Two iron spheres each have a volume of 10 cm^3 and a mass of 78g. How many electrons must be placed on each sphere so that the electrostatic force of repulsion between them just balances the force of gravitational attraction? Assume the distance between the spheres is more than 2 meters.

To calculate the number of electrons needed, we need to determine the electrostatic force of repulsion and the force of gravitational attraction between two iron spheres.

1. Calculate the mass of each sphere:
Given mass = 78g
Number of spheres = 2
Mass of each sphere = Total mass/Number of spheres
= 78g/2
= 39g

2. Convert the mass of each sphere to kilograms:
Mass in kilograms = Mass in grams/1000
= 39g/1000
= 0.039kg

3. Calculate the volume of each sphere in meters cubed (m^3):
Given volume = 10 cm^3
Since 1 cm^3 = 10^-6 m^3,
Volume in m^3 = Volume in cm^3 * (1 cm^3/10^-6 m^3)
= 10 cm^3 * 10^-6 m^3/cm^3
= 10 * 10^-6 m^3
= 10^-5 m^3

4. Calculate the radius of each sphere:
Volume of a sphere = (4/3) * π * r^3
10^-5 m^3 = (4/3) * π * r^3
r^3 = (3 * 10^-5 m^3) / (4/3 * π)
r = ∛((3 * 10^-5 m^3) / (4/3 * π))

5. Calculate the electrostatic force of repulsion between the spheres:
Coulomb's Law states that the force of electrostatic repulsion (Fe) between two charged objects is given by:
Fe = (k * q^2) / r^2,
where k is the electrostatic constant, q is the charge of each object, and r is the distance between them.

Since the distance is more than 2 meters, we can assume it is in the range ∞.

Therefore, Fe = (k * q^2) / r^2 = (k * q^2) / (∞)^2
= 0

Hence, the electrostatic force of repulsion when the distance is more than 2 meters is zero.

6. Calculate the force of gravitational attraction between the spheres:
The force of gravitational attraction (Fg) between two objects is given by:
Fg = (G * m1 * m2) / r^2,
where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between them.

The gravitational constant (G) = 6.67430 x 10^-11 N(m/kg)^2.

Hence, Fg = (6.67430 x 10^-11 N(m/kg)^2 * 0.039kg * 0.039kg) / (∞)^2
= 0

Therefore, the force of gravitational attraction when the distance is more than 2 meters is also zero.

Since both the electrostatic force of repulsion and the force of gravitational attraction are zero when the distance is more than 2 meters, no number of electrons can balance the forces.

To determine the number of electrons required on each iron sphere for the electrostatic force of repulsion to balance the force of gravitational attraction, we need to follow a few steps:

Step 1: Calculate the mass of each iron sphere.
Since both spheres have the same volume and mass, each sphere has a mass of 78g.

Step 2: Calculate the total charge required on each sphere.
The charge on an electron is -1.6 x 10^-19 Coulombs. We need to find the total charge required (in coulombs) to balance the gravitational force.
The gravitational force between two spheres is given by the formula: F = (G * m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (6.67430 x 10^-11 N.m^2/kg^2)
m1 and m2 are the masses of two spheres
r is the distance between the spheres

Since the gravitational force and the electrostatic force are inversely proportional to the square of the distance, we can assume the distance between the spheres is large enough that the gravitational force becomes negligible compared to the electrostatic force.

Step 3: Express the charge in terms of the number of electrons required.
The charge on a single electron is -1.6 x 10^-19 Coulombs, so we can calculate the number of electrons.

Now, let's perform the calculations.

Step 1: The mass of each iron sphere is 78g.

Step 2: Since the distance between the spheres is large, we can assume the gravitational force is negligible.

Step 3: To calculate the total charge required on each sphere, we use the formula:
F = (k * q1 * q2) / r^2

where F is the electrostatic force, k is the electrostatic constant (which equals 8.988 × 10^9 N m^2 C^-2), q1 and q2 are the charges on the two spheres, and r is the distance between them (which we assumed to be more than 2 meters).

Since we want the electrostatic force to balance the gravitational force (which is negligible here), we set the electrostatic force equal to zero:
0 = (k * q1 * q2) / r^2

Simplifying:
q1 * q2 = 0

To have a net zero charge, we need the charges on both spheres to be equal and opposite:
q1 = -q2

Substituting -q2 for q1:
(-q2) * q2 = 0

Simplifying:
q2^2 = 0

Taking the square root of both sides:
q2 = 0

Thus, we have a net zero charge on each sphere. Therefore, no electrons need to be placed on each sphere to balance the force of attraction.