Two point charges 0.600 m apart experience a repulsive force of 0.400 N. The sum of the two charges equals 1.00*10^-5 C. Find the values of the two charges.

To solve this problem, we can use Coulomb's Law, which states that the force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. Coulomb's Law equation is given by:

F = k * (|q1| * |q2|) / r^2

where F is the force, k is the electrostatic constant (9.0 x 10^9 N*m^2/C^2), q1 and q2 are the magnitudes of the two charges, and r is the distance between them.

Given that the force is 0.400 N, the distance is 0.600 m, and the sum of the charges is 1.00 * 10^-5 C, we can set up two equations using Coulomb's Law:

0.400 = k * (|q1| * |q2|) / (0.600)^2 (Equation 1)
1.00 * 10^-5 = |q1| + |q2| (Equation 2)

We know that the charges are repulsive, so they have the same sign (either positive or negative).

Let's solve Equation 2 for one charge in terms of the other:

|q1| = (1.00 * 10^-5) - |q2|

Now, substitute this expression for |q1| in Equation 1:

0.400 = k * ((1.00 * 10^-5) - |q2|) * |q2| / (0.600)^2

Simplifying:

0.400 = k * (1.00 * 10^-5 - |q2|) * |q2| / 0.360

Rearranging:

0.400 * 0.360 = k * (1.00 * 10^-5 - |q2|) * |q2|

0.144 = k * (1.00 * 10^-5 - |q2|) * |q2|

Now, substitute the value of k (9.0 x 10^9 N*m^2/C^2) and solve for |q2|:

0.144 = (9.0 x 10^9) * (1.00 * 10^-5 - |q2|) * |q2|

0.144 = 9.0 x 10^4 * (1.00 * 10^-5 - |q2|) * |q2|

0.144 = 9.0 x 10^4 * (1.00 * 10^-5 * |q2| - |q2|^2)

0.144 = 9.0 x 10^-1 * 1.00 * 10^-5 * |q2| - 9.0 x 10^4 * |q2|^2

0.144 = 9.0 * |q2| - 9.0 x 10^4 * |q2|^2

Now, we have a quadratic equation. Rearrange it in standard form:

9.0 x 10^4 * |q2|^2 - 9.0 * |q2| + 0.144 = 0

Solve this quadratic equation using any suitable method, such as factoring, completing the square, or using the quadratic formula. The solutions for |q2| will give us the values of the two charges.