Two point charges are 10 cm apart and their combined charge is 9μC. Ifthey repel each other with a force of 18N, then calculate the magnitude ofeach charge
Question
Two point charges are 10 cm apart and their combined charge is 9μC. Ifthey repel each other with a force of 18N, then calculate the magnitude ofeach charge
Solution
Sure, let's solve this step by step.
We know that the force between two charges is given by Coulomb's law:
F = k * |q1*q2| / r^2
where: F is the force between the charges, k is Coulomb's constant (9 * 10^9 N m^2/C^2), q1 and q2 are the charges, r is the distance between the charges.
We are given: F = 18 N, r = 10 cm = 0.1 m, q1 + q2 = 9 μC = 9 * 10^-6 C.
We can rearrange Coulomb's law to find q1*q2:
q1*q2 = F * r^2 / k = 18 N * (0.1 m)^2 / (9 * 10^9 N m^2/C^2) = 0.2 * 10^-6 C^2.
Now, we have a system of two equations:
q1 + q2 = 9 * 10^-6 C, q1*q2 = 0.2 * 10^-6 C^2.
This is a quadratic equation in disguise. If we let q1 = x and q2 = 9*10^-6 - x, we can substitute q2 into the second equation to get:
x * (9*10^-6 - x) = 0.2 * 10^-6, 9x - x^2 = 0.2, x^2 - 9x + 0.2 = 0.
Solving this quadratic equation will give us the values of q1 and q2. The solutions to this equation are x = 4.5 * 10^-6 C and x = 4.5 * 10^-6 C. Therefore, the two charges are each 4.5 μC.
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