Two small, conducting spheres, initially uncharged, are brought into contact and given a total charge of 5.00x10^4 electrons. The spheres are then separated until their centers are 12.0 cm apart. Assuming the total number of electrons on the spheres remains the same as they are separated: a) What is the magnitude of the force that each sphere exerts on the other? b) Is the force attractive or repulsive?
Answered on
a) To find the magnitude of the force that each sphere exerts on the other, we can start by calculating the amount of charge given to the spheres. The total charge is the charge of one electron multiplied by the total number of electrons:
Charge of one electron, e = -1.60 x 10^-19 C (Coulombs)
Total charge, Q = (5.00 x 10^4 electrons) x (-1.60 x 10^-19 C/electron) Q = -8.00 x 10^-15 C
Since the spheres are identical and were in contact when they were charged, the charge is distributed equally between them. Therefore, each sphere acquires half of the total charge:
Charge per sphere, q = Q / 2 q = (-8.00 x 10^-15 C) / 2 q = -4.00 x 10^-15 C
We drop the negative sign because we are only interested in the magnitude of the charge for calculating forces.
Next, we will use Coulomb's Law to find the force between the two charged spheres. Coulomb's Law is given by:
F = k * |q1 * q2| / r^2
where: F is the magnitude of the force between the spheres, k is Coulomb's constant (8.9875 x 10^9 N·m²/C²), q1 and q2 are the magnitudes of the charges, r is the distance between the centers of the two charges.
Since the spheres have the same charge, q1 = q2 = -4.00 x 10^-15 C, and the distance r is 12.0 cm or 0.12 meters:
F = (8.9875 x 10^9 N·m²/C²) * |(-4.00 x 10^-15 C) * (-4.00 x 10^-15 C)| / (0.12 m)^2
F = (8.9875 x 10^9 N·m²/C²) * (16.00 x 10^-30 C²) / (0.0144 m²)
F ≈ (8.9875 x 10^9 N·m²/C²) * (1.1111 x 10^-27 C²/m²)
F ≈ 1.0002 x 10^-17 N
Thus, the magnitude of the force that each sphere exerts on the other is approximately 1.00 x 10^-17 N.
b) Since both spheres are negatively charged (as they both received electrons), the force between them would be repulsive. Like charges repel each other according to the principles of electrostatics.