Finding an unknown charge in a system of discrete point charges.

[Zift]

A point particle that has a charge of 15.0 µC is located at x = 0, y = 0 (we'll call it q₀ and a point particle that has a charge q is located at x = 12.0 cm, y = 0 (we'll call it q₁. The electric force on a point particle that has a charge of 6.0 µC at x = 24.0 cm, y = 0 is -(19.7) N $$\hat{i}$$ (we'll call it q₂. When this 6.0-µC charge is repositioned at x = 17.8 cm, y = 0, the electric force on it is zero. Determine the charge q.

First I tried to equate the electric force on q₂ to the vector of the forces on q₂ due to q₀ and q₁ using coulomb's law and then solve for q₁, but this does not yield the correct value. I have been able to do all the other problems in the section on discrete charge distributions, but I'm completely stumped on this one. I also tried to do the same procedure but with the electric force on q₂ equal to zero as indicated in the last sentence and but its yields a different (and still incorrect) value. what am I doing wrong. Am i missing something? Is there crucial information I should be deriving from the last sentence?

 

[ehild]

This problem looks over-defined, and I also found the results controversial.

ehild

 

[kerimek]

I would approach this problem by first identifying all of the known variables and equations that can be used to solve for the unknown charge q. In this case, we have three point charges (q0, q1, and q2) with known positions and charges, and an electric force acting on q2 at two different positions. We can use Coulomb's law to calculate the electric force between two point charges, and we also know that the electric force is a vector quantity.

From the given information, we can set up the following equations:

1) Electric force on q2 at x = 24.0 cm, y = 0: F = k(q2)(q0)/r^2 + k(q2)(q1)/(24.0 cm)^2

2) Electric force on q2 at x = 17.8 cm, y = 0: F = k(q2)(q0)/r^2 + k(q2)(q1)/(17.8 cm)^2

We also know that the electric force on q2 at x = 17.8 cm, y = 0 is zero, so we can set the two equations equal to each other and solve for q1:

k(q2)(q0)/r^2 + k(q2)(q1)/(24.0 cm)^2 = k(q2)(q0)/r^2 + k(q2)(q1)/(17.8 cm)^2

Simplifying and solving for q1, we get:

q1 = (24.0 cm)^2/(17.8 cm)^2 * q0

Plugging in the known values for q0 (15.0 µC) and q1 (6.0 µC), we can solve for q2:

q2 = (17.8 cm)^2/(24.0 cm)^2 * (15.0 µC)/(6.0 µC) = 10.5 µC

Therefore, the unknown charge q is 10.5 µC.

It is important to note that in this problem, we are assuming that the system of discrete point charges is in equilibrium, meaning that the net force on any point charge is zero. This assumption is crucial in solving the problem and may be the reason why your previous attempts did not yield the correct value for q. Additionally, the last sentence in the problem statement (stating that the electric force