Question

You have two small spheres, each with a mass of 2.40 grams, separated by a distance of 10.0 cm. You remove the same number of electrons from each sphere.

1) What is the charge on each sphere if their gravitational attraction is exactly equal to their electrical repulsion?

2) How many electrons did you remove from each sphere?

Answer 1

q = 2.066* 10⁻¹³ C.

n = 1,291,250 electrons.

Step-by-step explanation:

1)

• If the gravitational attraction is equal to their electrical repulsion, we can write the following equation:

`F_(g) = F_(c) (1)`

• where Fg is the gravitational attraction, that can be written as follows according Newton's Universal Law of Gravitation:

`F_(g) = G*(m_(1)*m_(2))/(r_(12)^(2)) (2)`

• Fc, due to it is the electrical repulsion between both charged spheres, must obey Coulomb's Law (assuming we can treat both spheres as point charges), as follows:

`F_(c) = k*(q_(1)*q_(2))/(r_(12)^(2)) (3)`

• since m₁ = m₂ = 0.0024 kg, and r₁₂ = 0.1m, G and k universal constants, and q₁ = q₂ = Q, we can replace the values in (2) and (3), so we can rewrite (1) as follows:

`G*((0.0024kg)^(2))/(r_(12)^(2)) = k*(Q^(2))/(r_(12)^(2)) (4)`

• Since obviously the distance is the same on both sides, we can cancel them out, and solve (4) for Q² first, as follows:

`Q^(2) = (6.67e-11*(0.0024kg)^(2))/(9e9Nm2/C2) = 4.27*e-26 C2 (5)`

• Since both charges are the same, the charge on each sphere is just the square root of (5):
• Q = 2.066* 10⁻¹³ C.

2)

• Assuming that both spheres were electrically neutral before being charged, the negative charge removed must be equal to the positive charge on the spheres.
• Now, since each electron carries an elementary charge equal to -1.6*10⁻¹⁹ C, in order to get the number of electrons removed from each sphere, we need to divide the charge removed from each sphere (the outcome of part 1) with negative sign) by the elementary charge, as follows:
• 
  `n_(e) =(-2.066e-13C)/(-1.6e-19C) = 1,291,250 electrons. (6)`