Two identical positive charges are placed 15 cm apart and the magnitude of the electric force between them is 5.4 Newtons. They then exchange charge (one charge loses some electrons and the other gains the same amount of electrons), but they both still remain positive. After they have exchanged charge and are placed 15 cm apart the magnitude of the electric force is now 3.8 Newtons. How much charge, in micro-coulombs, was exchanged? Use the absolute value of the charge exchanged so the answer is positive.
To find the amount of charge exchanged between the two identical charges, we can use Coulomb's law, which relates the magnitude of the electric force to the charges and distance. The equation for Coulomb's law is:
F = k * (q1 * q2) / r^2
where F is the magnitude of the electric force, k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between the charges.
In the initial scenario, when the charges were 15 cm apart, we have:
5.4 N = k * (q1 * q2) / (0.15 m)^2
In the final scenario, after the charge exchange, we have:
3.8 N = k * (q2 * q2) / (0.15 m)^2
We can divide the two equations to eliminate Coulomb's constant and the distance factor:
5.4 N / 3.8 N = (q1 * q2) / (q2 * q2)
Simplifying, we get:
1.421 N = q1 / q2
Now, since both charges remained positive after the exchange, we can take the ratio of the magnitudes of the charges:
1.421 = |q1| / |q2|
Assuming |q2| > |q1| (since one charge lost electrons and the other gained the same amount of electrons), we can rewrite the equation as:
1.421 = q1 / |q2|
Solving for |q1|, we get:
|q1| = 1.421 * |q2|
Now, we can substitute the values from the first equation:
5.4 N = k * (q1 * q2) / (0.15 m)^2
To get rid of Coulomb's constant, we can divide both sides by k and multiply by (0.15 m)^2:
(5.4 N * (0.15 m)^2) / k = q1 * q2
Since q1 = 1.421 * |q2|, we can substitute it in and rearrange the equation:
(5.4 N * (0.15 m)^2) / k = (1.421 * |q2|) * q2
Simplifying further:
|q2|^2 = (5.4 N * (0.15 m)^2) / (1.421 * k)
Now, we can calculate the magnitude of |q2|:
|q2| = sqrt[(5.4 N * (0.15 m)^2) / (1.421 * k)]
Using the value of Coulomb's constant, k = 8.99 x 10^9 N m^2 / C^2, we can substitute it in:
|q2| = sqrt[(5.4 N * (0.15 m)^2) / (1.421 * (8.99 x 10^9 N m^2 / C^2))]
Evaluating this expression, we find:
|q2| ≈ 2.911 x 10^-6 C
Since the charges exchanged were of equal magnitude, we can conclude that the magnitude of the charge exchanged is:
|q1| ≈ 1.421 * 2.911 x 10^-6 C ≈ 4.140 x 10^-6 C
Therefore, approximately 4.140 micro-coulombs of charge were exchanged.