Two identical charges repel each other with a force 0.1mN and are moved additional 5.0cm a part the repulsive force is reduced to 25 x 10^-6N, (a) How far were they originally (b) what is the size of the charges
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electrical force (F) is proportional to electric field (E) and E is proportional to inverse square of the distance (1/r^2) between charges.
Therefore, F ~ 1/r^2.
Now compare the two force given F1 and F2 then
F1/F2 = r2^2 / r1^2
To solve this problem, we can use Coulomb's law, which states that the electric force between two charges is inversely proportional to the square of the distance between them. It can be mathematically written as:
F = k * (q1 * q2) / r^2
where F is the electric force, k is Coulomb's constant (9 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
(a) To determine the original distance between the charges, we can set up an equation using the given information.
Given:
Initial force (F1) = 0.1 mN = 0.1 x 10^-3 N
Final force (F2) = 25 x 10^-6 N
Initial distance (r1) = unknown
Final distance (r2) = 5.0 cm = 5.0 x 10^-2 m
Using Coulomb's law, we have two equations:
F1 = k * (q1 * q2) / r1^2
F2 = k * (q1 * q2) / r2^2
Dividing the above two equations eliminates the charged terms:
F1 / F2 = (r2^2 / r1^2)
Substituting the given values:
(0.1 x 10^-3 N) / (25 x 10^-6 N) = (5.0 x 10^-2 m)^2 / r1^2
Now, solve for r1:
r1^2 = (5.0 x 10^-2 m)^2 * (25 x 10^-6 N) / (0.1 x 10^-3 N)
r1^2 = (5.0 x 10^-2 m)^2 * (25 x 10^-6 N) / (0.1 x 10^-3 N)
r1^2 = 0.0125 m^2
r1 = sqrt(0.0125 m^2)
r1 = 0.112 m
Therefore, the two charges were originally 0.112 meters apart.
(b) To find the size of the charges, we can rearrange Coulomb's law equation:
F = k * (q1 * q2) / r^2
Solving for the product of the charges (q1 * q2):
(q1 * q2) = (F * r^2) / k
Using the initial force (F1) and distance (r1) values:
(q1 * q2) = (0.1 x 10^-3 N) * (0.112 m)^2 / (9 x 10^9 N m^2/C^2)
Simplifying:
(q1 * q2) = 1.25 x 10^-14 C^2
Since the charges are identical, q1 = q2 = q.
Thus, we have:
q^2 = 1.25 x 10^-14 C^2
Taking the square root to find q:
q = sqrt(1.25 x 10^-14 C^2)
q = 1.118 x 10^-7 C
Therefore, the size of the charges is approximately 1.118 x 10^-7 C.