Answer: To find the charge on the fixed sphere, we can use Coulomb's Law, which states that the electrostatic force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
The formula for the electrostatic force (F) between two charged objects is given by:
F = k * (|q1 * q2|) / r^2
where:
F = electrostatic force
k = Coulomb's constant (approximately 8.99 × 10^9 N m²/C²)
q1 and q2 = charges of the two objects (in Coulombs)
r = distance between the centers of the two spheres (in meters)
Given:
Mass of the suspended sphere (m) = 75 g = 0.075 kg
Charge of the suspended sphere (q1) = 4.0 MC = 4.0 × 10^6 C
Now, the first thing we need to do is find the charge (q2) on the fixed sphere. For this, we need to measure the force of repulsion between the two spheres.
The gravitational force (F_gravity) acting on the suspended sphere is given by:
F_gravity = m * g
where g is the acceleration due to gravity (approximately 9.81 m/s²).
The electrostatic force (F_electric) between the two spheres causes the repulsion. We can equate the electrostatic force to the gravitational force:
F_electric = F_gravity
Now, let's solve for the charge (q2) on the fixed sphere:
k * (|q1 * q2|) / r^2 = m * g
Substitute the known values:
8.99 × 10^9 N m²/C² * (|4.0 × 10^6 C * q2|) / r^2 = 0.075 kg * 9.81 m/s²
Now, rearrange the equation to solve for q2:
|4.0 × 10^6 C * q2| = (0.075 kg * 9.81 m/s² * r^2) / 8.99 × 10^9 N m²/C²
|4.0 × 10^6 C * q2| = (0.073575 kg m² / s²) / 8.99 × 10^9 N m²/C²
|4.0 × 10^6 C * q2| ≈ 8.1846 × 10^-12 C
Now, find the magnitude of q2:
q2 = (8.1846 × 10^-12 C) / 4.0 × 10^6 C
q2 ≈ 2.04615 × 10^-18 C
Finally, since the suspended sphere is repelled, the charge on the fixed sphere (q2) is positive.
Therefore, the charge on the fixed sphere is approximately 2.04615 × 10^-18 Coulombs.