Two charges are separated until they are a distance of 191 cm apart, causing the electric force between them to increase by a factor of exactly 5. What was their initial separation distance?
Let's first assume that the initial separation distance is "x".
According to the question, when the charges are separated by a distance of x, the electric force between them is F.
When the charges are separated by a distance of 191 cm, the electric force between them increases by a factor of 5.
So, the new electric force is 5F.
We can use Coulomb's law to relate the initial and final force with the separation distance:
F = k * (q1 * q2) / (x^2)
5F = k * (q1 * q2) / (191^2)
Dividing the two equations:
5 = (k * (q1 * q2) / (191^2)) / (k * (q1 * q2) / (x^2))
Simplifying:
5 = x^2 / 191^2
Taking the square root of both sides:
√5 = x / 191
x = 191 * √5
Hence, the initial separation distance is approximately 191 * √5 cm.
To solve this problem, we can use the formula for electric force between two charges:
Electric force (F) = k * (q1 * q2) / r^2
Where:
F is the electric force,
k is the electrostatic constant (9 * 10^9 N.m^2/C^2),
q1 and q2 are the magnitudes of the charges, and
r is the separation distance between the charges.
In the given problem, we are told that the electric force increases by a factor of 5 when the separation distance decreases. Let's denote the initial separation distance as r1 and the final separation distance as r2.
We can set up the following equation based on the given information:
F2 = 5 * F1, where F1 is the initial electric force and F2 is the final electric force.
Since F2 = (k * (q1 * q2)) / r2^2 and F1 = (k * (q1 * q2)) / r1^2, we can rewrite the equation as:
(k * (q1 * q2)) / r2^2 = 5 * (k * (q1 * q2)) / r1^2
Now, we can simplify the equation:
r1^2 = 5 * r2^2
Taking the square root of both sides gives:
r1 = sqrt(5) * r2
We are given that r2 is 191 cm, so substituting the value:
r1 = sqrt(5) * 191 cm
Now we can calculate the value of r1:
r1 = sqrt(5) * 191 cm ≈ 303.83 cm
Therefore, the initial separation distance between the charges was approximately 303.83 cm.