Two point charges totaling 9.00 µC exert a repulsive force of 0.300 N on one another when separated by 0.596 m. (A) What is the charge on each? (B) What is the charge on each if the force is attractive?
smallest charg
To find the charge on each point charge, we can use Coulomb's law, which states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The formula for Coulomb's law is:
F = k * (|q1| * |q2|) / r^2
Where:
- F is the force between the charges,
- k is Coulomb's constant (k = 8.99 x 10^9 N·m^2/C^2),
- q1 and q2 are the charges on the two point charges,
- r is the distance between the charges.
Let's solve the problem step by step:
(A) When the force is repulsive:
We are given:
- Total charge = 9.00 µC = 9.00 x 10^-6 C
- Force F = 0.300 N
- Distance r = 0.596 m
Using Coulomb's law, we can rearrange the formula to solve for the charges:
F * r^2 = k * (|q1| * |q2|)
(|q1| * |q2|) = (F * r^2) / k
Substituting the given values:
(|q1| * |q2|) = (0.300 N * (0.596 m)^2) / (8.99 x 10^9 N·m^2/C^2)
(|q1| * |q2|) = 0.0126 C^2
Since the charges are repulsive, they have the same sign. Let's call the charge on each point charge q:
q * q = 0.0126 C^2
Taking the square root of both sides, we have:
q = sqrt(0.0126 C^2) = 0.112 C ≈ 1.12 x 10^-1 C
Therefore, each point charge has a charge of approximately 1.12 x 10^-1 C.
(B) When the force is attractive:
Using the same formula and steps as in part (A), we substitute the given values:
Total charge = 9.00 µC = 9.00 x 10^-6 C
Force F = 0.300 N
Distance r = 0.596 m
(|q1| * |q2|) = (0.300 N * (0.596 m)^2) / (8.99 x 10^9 N·m^2/C^2)
(|q1| * |q2|) = 0.0126 C^2
Since the charges are now attractive, they have opposite signs. Let's assume one charge is positive (q1) and the other is negative (q2):
q1 * q2 = -0.0126 C^2
To find the smallest charges, we take the square root of the absolute value of the product:
q = sqrt(|q1 * q2|) = sqrt(0.0126 C^2) = 0.112 C ≈ 1.12 x 10^-1 C
Therefore, the smallest charge on each point charge when the force is attractive is also approximately 1.12 x 10^-1 C.