(1 point) The sum of two point charges is +8μC. When they are 3.7cm apart, each experiences a force of 63 N
. Find the charges given that the force is:
a) repulsive. (List your two answers in increasing order of magnitude) μC
b) attractive. (List your two answers in increasing order of magnitude) μC
To find the charges given the force between two point charges, we can use Coulomb's law:
F = k * (q1 * q2) / r^2
where F is the force between the charges, q1 and q2 are the magnitudes of the charges, r is the distance between the charges, and k is the electrostatic constant.
Given that the force is 63 N and the distance between the charges is 3.7 cm, we have:
F = 63 N
r = 3.7 cm = 0.037 m
k = 9.0 x 10^9 N m^2/C^2 (electrostatic constant)
a) For repulsive charges, the force and the charges have the same sign (positive or negative). Let's assume the charges are q1 and q2, with q1 > q2.
Since the force is repulsive, q1 and q2 are both positive charges.
Using Coulomb's law, we can rearrange the formula to solve for the charges:
F = k * (q1 * q2) / r^2
63 N = (9.0 x 10^9 N m^2/C^2) * (q1 * q2) / (0.037 m)^2
Rearranging this equation to solve for q1:
q1 = (F * (r^2)) / (k * q2)
= (63 N * (0.037 m)^2) / (9.0 x 10^9 N m^2/C^2 * q2)
= 9.1 x 10^-5 C / q2
We can now plug in different values for q2 and solve for q1:
If q2 = 1 μC, then q1 = 9.1 x 10^-5 C / 1 μC = 9.1 x 10^-5 C / 10^-6 C = 91 C
If q2 = 2 μC, then q1 = 9.1 x 10^-5 C / 2 μC = 9.1 x 10^-5 C / 2 x 10^-6 C = 45.5 C
So the charges in increasing order of magnitude for repulsive forces are: 1 μC, 2 μC.
b) For attractive charges, the force and the charges have opposite signs. Let's assume the charges are q1 and q2, with q1 > q2.
Since the force is attractive, q1 is positive and q2 is negative.
Using Coulomb's law, we can rearrange the formula to solve for the charges:
F = k * (q1 * q2) / r^2
63 N = (9.0 x 10^9 N m^2/C^2) * (q1 * (-q2)) / (0.037 m)^2
Rearranging this equation to solve for q1:
q1 = -(F * (r^2)) / (k * q2)
= -(63 N * (0.037 m)^2) / (9.0 x 10^9 N m^2/C^2 * q2)
= -9.1 x 10^-5 C / q2
We can now plug in different values for q2 and solve for q1:
If q2 = 1 μC, then q1 = -9.1 x 10^-5 C / 1 μC = -9.1 x 10^-5 C / 10^-6 C = -91 C
If q2 = 2 μC, then q1 = -9.1 x 10^-5 C / 2 μC = -9.1 x 10^-5 C / 2 x 10^-6 C = -45.5 C
So the charges in increasing order of magnitude for attractive forces are: -2 μC, -1 μC.
To find the charges given the force and their separation, we can apply Coulomb's Law, which states that the force between two point charges can be calculated using the equation:
F = k * (q1 * q2) / r^2
where F is the force, k is the electrostatic constant (9 × 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the separation between the charges.
Let's solve for each case separately:
a) Repulsive force:
Since the force is repulsive, the two charges must have the same sign, either positive or negative. Let's assume they are both positive.
We are given:
F = 63 N
r = 0.037 m (3.7 cm)
k = 9 × 10^9 N m^2/C^2
Using Coulomb's Law, we can rearrange the equation to solve for the product of the charges:
q1 * q2 = (F * r^2) / k
q1 * q2 = (63 N * (0.037 m)^2) / (9 × 10^9 N m^2/C^2)
q1 * q2 = 9.468 * 10^(-8) C^2
To find the charges, we need to factorize 9.468 * 10^(-8) into two numbers that multiply to give this value and then convert them to microcoulombs.
Possible factorizations include:
1 * 9.468 × 10^(-8) μC
2 * 4.734 × 10^(-8) μC
3 * 3.156 × 10^(-8) μC
...
The two answers in increasing order of magnitude for the charges would be:
1 μC and 9.468 × 10^(-8) μC
b) Attractive force:
Since the force is attractive, the two charges must have opposite signs, one positive and one negative. Let's assume q1 is positive and q2 is negative.
Using the same values for F, r, and k as in the previous case, we can again rearrange Coulomb's Law to solve for the product of the charges:
q1 * q2 = (F * r^2) / k
q1 * q2 = (63 N * (0.037 m)^2) / (9 × 10^9 N m^2/C^2)
q1 * q2 = 9.468 * 10^(-8) C^2
We can then factorize 9.468 * 10^(-8) as before, but this time with one charge positive and the other negative.
The two answers in increasing order of magnitude for the charges would be:
-1 μC and 9.468 × 10^(-8) μC