Two spherical objects are separated by a distance of 2.41 x 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 4.18 x 10-21 N. How many electrons did it take to produce the charge on one of the objects?
Can you help at all? Please give me an idea where to start. Thank you.
F = Q^2 / (4 pi eps d^2)
find Q
then the number electrons is
N = Q /e
where e is the electron's charge (look it up)
The distance between them is
d = 2.41 x 10-3 m
The charge on each sphere is Q
The force of each sphere on the other is
F = k Q^2/d^2 = 4.18 x 10-21 N
Look up the Coulomb's Law constant k and solve that equation for Q.
The number of electrons is
N = Q/e
where e is the electron charge.
Ok. I did that and I found that Q= 1.64 x 10 ^ -18.
I then divided that by 1.6 x 10 ^ -19.
The answer I got was 10.23.
I am being told that is the wrong answer. Could you please double check that for me?
Thank you!!
To find the number of electrons required to produce the charge on one of the objects, we can use the equation for the electrostatic force between charged objects.
The electrostatic force between two charged objects is given by Coulomb's Law:
F = k * (q1 * q2) / r^2
Where:
F is the electrostatic force
k is the electrostatic constant (9 x 10^9 N m^2/C^2)
q1 and q2 are the magnitudes of the charges of the two objects
r is the separation distance between the objects
In this case, both objects have acquired the same negative charge, so their charges are equal in magnitude. Let's say the charge on each object is q.
Since each object experiences the same electrostatic force, we can write:
F = F1 = F2 = k * (q * q) / r^2
Given that F = 4.18 x 10^-21 N and r = 2.41 x 10^-3 m, we can rearrange the equation to solve for q:
q^2 = (F * r^2) / k
q^2 = (4.18 x 10^-21 N * (2.41 x 10^-3 m)^2) / (9 x 10^9 N m^2/C^2)
Now, we can calculate q:
q^2 = (4.18 x 10^-21 N * 5.8081 x 10^-9 m^2) / (9 x 10^9 N m^2/C^2)
q^2 = 2.52 x 10^-29 C^2/C^2
q = √(2.52 x 10^-29 C^2)
q ≈ 1.59 x 10^-15 C
We know that one electron has a charge of approximately -1.6 x 10^-19 C. So, we can find the number of electrons required by dividing the charge on one object by the charge of one electron:
Number of electrons = q / (-1.6 x 10^-19 C)
Number of electrons ≈ (1.59 x 10^-15 C) / (-1.6 x 10^-19 C)
Number of electrons ≈ -9.94 x 10^3
Since we need a positive value for the number of electrons, we take the absolute value:
Number of electrons ≈ 9.94 x 10^3
Therefore, it took approximately 9.94 x 10^3 electrons to produce the charge on one of the objects.