two spherical objects are separated by a distance of 9x10^-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 4.55 x 10^-21 N. How many electrons did it take to produce the charge on one of the objects?
the answer is 40 electrons but i have no idea how to do it
From the force, and coulomb's force equation, solve for q^2 then q.
divide q by e, the charge on one electron to get the number of electrons.
To determine the number of electrons required to produce the charge on one of the objects, we can use Coulomb's law and the fundamental unit of charge.
Step 1: Recall Coulomb's Law
Coulomb's law states that the magnitude of 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. Mathematically, it is expressed as:
F = k*q1*q2 / r^2
Where F is the magnitude of the electrostatic force, k is the electrostatic constant (9 x 10^9 N*m^2/C^2), q1 and q2 are the charges, and r is the distance between the objects.
Step 2: Determine the charge on one of the objects
Since both objects acquired the same negative charge due to the addition of electrons, we can assume that the charges on each object are equal. Let's denote this charge as q.
Step 3: Set up the equation
From the given information, we have:
F = 4.55 x 10^-21 N
r = 9 x 10^-3 m
k = 9 x 10^9 N*m^2/C^2
Plugging these values into Coulomb's law, we get:
4.55 x 10^-21 N = (9 x 10^9 N*m^2/C^2) * q^2 / (9 x 10^-3 m)^2
Simplifying the equation, we have:
4.55 x 10^-21 * (9 x 10^-3)^2 = 9 x 10^9 * q^2
Step 4: Solve for q (charge on one object)
q^2 = (4.55 x 10^-21 * (9 x 10^-3)^2) / (9 x 10^9)
q^2 = 4.55 x 10^-39 / 9
q^2 = 5.06 x 10^-40
q ≈ 7.12 x 10^-20 C
Step 5: Determine the number of electrons
We know that 1 electron has a charge of approximately -1.6 x 10^-19 C. Therefore, to find the number of electrons, we divide the charge on one object by the charge of a single electron:
Number of electrons = q / (charge of one electron)
Number of electrons = (7.12 x 10^-20 C) / (-1.6 x 10^-19 C)
Number of electrons ≈ 40
Therefore, it took approximately 40 electrons to produce the negative charge on one of the objects.