Two spherical objects are separated by a distance of 2.70 × 10-3 m. The objects are initially electrically neutral and are very small compared to the distance between them. Each object acquires the same negative charge due to the addition of electrons. As a result, each object experiences an electrostatic force that has a magnitude of 2.02 × 10-21 N. How many electrons did it take to produce the charge on one of the objects?
To determine how many electrons it took to produce the charge on one of the objects, we can use the formulas and principles related to electric charge and Coulomb's law.
Let's break down the problem step by step:
1. Electric force between two charged objects:
According to Coulomb's Law, the electrostatic force between two charged objects is given by:
F = k * (q1 * q2) / r^2
Where:
F is the electrostatic force
k is the electrostatic constant (8.99 × 10^9 N*m^2/C^2)
q1 and q2 are the charges on the objects
r is the distance between the objects.
2. Electric charge of an object:
The electric charge on an object is usually the result of an excess or deficiency of electrons. Electrons have a charge of -1.6 × 10^-19 C.
3. Equating the electrostatic force with the electric charge:
Based on the problem, the objects acquire the same negative charge, so let's assume the charge on one of the objects as q1 = -Q, and the charge on the other object as q2 = -Q. The negative sign indicates that the objects have acquired a negative charge.
The distance between the objects, r, is given as 2.70 × 10^-3 m, and the electrostatic force, F, is given as 2.02 × 10^-21 N.
By substituting these values into Coulomb's Law, we get the equation:
2.02 × 10^-21 = (8.99 × 10^9 * (-Q) * (-Q)) / (2.70 × 10^-3)^2
Simplifying and solving for Q:
Q^2 = (2.02 × 10^-21 * (2.70 × 10^-3)^2) / (8.99 × 10^9)
Q^2 ≈ 0.0000162
Taking the square root of both sides:
Q ≈ √0.0000162
Q ≈ 0.00403 C
4. Calculating the number of electrons:
To determine the number of electrons that contributed to the charge on one of the objects, we can divide the charge Q by the charge of a single electron (-1.6 × 10^-19 C):
Number of electrons = Q / (-1.6 × 10^-19)
Number of electrons ≈ 0.00403 C / (-1.6 × 10^-19 C)
Number of electrons ≈ -2.52 × 10^16
Therefore, it took approximately 2.52 × 10^16 electrons to produce the charge on one of the objects.