Two identical point charges repel each other with a force of 4N. When the charges are moved 5mm further apart, the resultant force is reduced to 1N.
How far apart were the charges originally?
moving .005m force is 1/4 original. sounds like the distance was doubled, or original distance was .005m Remember force is proportional to 1/d^2
To solve this problem, 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.
Let's denote the original distance between the charges as d.
According to the problem, when the charges are at a distance of d, the resulting force is 4N. When they are moved 5mm further apart (at a distance of d + 5mm), the resulting force is reduced to 1N.
Using Coulomb's law, we can set up the following equations:
Original Force: k * q^2 / (d^2) = 4N ----(1)
New Force: k * q^2 / (d + 5mm)^2 = 1N ----(2)
Since the charges are identical, we can say that the two point charges have the same magnitude, so we can replace q^2 with q in our equations.
Dividing equation (1) by equation (2), we get:
(4N) / (1N) = (k * q / d^2) / (k * q / (d + 5mm)^2)
Simplifying the equation:
4 = (d + 5mm)^2 / d^2
Expanding (d + 5mm)^2:
4 = (d^2 + 10mm * d + 25mm^2) / d^2
Multiplying both sides by d^2:
4d^2 = d^2 + 10mm * d + 25mm^2
Subtracting d^2 from both sides:
3d^2 = 10mm * d + 25mm^2
Rearranging the equation:
3d^2 - 10mm * d - 25mm^2 = 0
This is a quadratic equation. We can solve it using the quadratic formula:
d = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation:
a = 3, b = -10mm, c = -25mm^2
Plugging in the values:
d = (-(-10mm) ± sqrt((-10mm)^2 - 4 * 3 * (-25mm^2))) / (2 * 3)
Simplifying:
d = (10mm ± sqrt(100mm^2 + 300mm^2)) / 6
d = (10mm ± sqrt(400mm^2)) / 6
d = (10mm ± 20mm) / 6
Now, we have two possible distances:
d1 = (10mm + 20mm) / 6 = 5mm
d2 = (10mm - 20mm) / 6 = -1.66mm
We can ignore the negative value, so the original distance between the charges was 5mm.
To find out how far apart the charges were originally, 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.
Let's assume the charges are q1 and q2, and the distance between them originally is d1.
Given:
Force when charges are originally separated by distance d1: F1 = 4N
Force when charges are separated by distance d2 (5mm further apart): F2 = 1N
Using Coulomb's Law, we can write the equation as:
F = k * (q1*q2) / d^2
Where:
F is the force between the charges
k is the Coulomb's constant (k ≈ 9 × 10^9 Nm^2/C^2)
d is the distance between the charges
Let's set up the equation for the two scenarios:
For the original configuration (F1 = 4N):
4N = k * (q1*q2) / d1^2
For the new configuration (F2 = 1N, d2 = d1 + 5mm):
1N = k * (q1*q2) / (d1 + 0.005m)^2
Now we have a system of equations with two variables (q1 and q2) and two equations. To solve it, we divide the second equation by the first equation:
(1N / 4N) = [(k * (q1*q2) / (d1 + 0.005m)^2)] / [(k * (q1*q2) / d1^2)]
1/4 = (d1^2) / (d1 + 0.005m)^2
After cross-multiplication and simplification, we get:
4 * (d1 + 0.005m)^2 = d1^2
Expanding and rearranging the equation:
4 * (d1^2 + 0.01d1 + 0.000025m^2) = d1^2
4d1^2 + 0.04d1 + 0.0001 = d1^2
3d1^2 - 0.04d1 - 0.0001 = 0
Now that we have a quadratic equation, we can solve for d1 using the quadratic formula:
d1 = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation, a = 3, b = -0.04, and c = -0.0001. Plugging these values into the quadratic formula, we can calculate d1.