1. ## More on Charges

Four particles form a square. The charges are $\displaystyle q_1 = q_4 = Q$ and $\displaystyle q_2 = q_3 = q$.

(a) What is $\displaystyle \frac Qq$ if the net electrostatic force on particles 1 and 4 is zero?

(b) Is there any value of q that makes the net electrostatic force on each of the four particles zero? Explain.

2. If I understand well how you draw the diagram then
$\displaystyle F_{EQ}=\frac{kQ^2}{(\sqrt{2}c)^2}$
$\displaystyle F_{Eq}=\frac{kqQ}{c^2}$ where c is one side of the square.
The forces are balanced.
In the direction of Q $\displaystyle F_{EQ}=2F_{Eq} cos(\frac{\pi}{4})$
I think you should get $\displaystyle \frac{Q}{q}= 2\sqrt{2}$

3. If you don't mind me asking, where did the $\displaystyle (\sqrt{2}c)^2$ come from?

How did you derive the force equations for Q and q?

4. $\displaystyle \sqrt{2}c$ is the diagonal of the square.
$\displaystyle F_{EQ}=\frac{kq_1q_2}{r^2}$ is the electric force equation between two charges.

5. Ok, I understand the diagonal.

But I guess I should rephrase my second question.

Why is it $\displaystyle Q^2$ for the force on Q, and $\displaystyle qQ$ for the force on q?

6. The electric force between two particles is proportional to each of the charges and inversely proportional to the square of the distance between them
$\displaystyle F_{EQ}=\frac{kq_1q_2}{r^2}$
The force of q1 on q2 is the same as q2 on q1. This is action-reaction principle.
Think of it as gravitation. It is the same. Mass is gravitational charge and "charge" is electric charge. The force of the moon on the earth is the same as the earth on the moon. Effects are different because mass of each is different.
So the force of Q on Q is $\displaystyle F_{EQ}=\frac{kQ^2}{r^2}$ and the force of q on q is $\displaystyle F_{EQ}=\frac{kqQ}{r^2}$ Notice that when the two charge are the same sign they tend to repulse each other. The opposite happen when they are of different sign. Here, Q and Q are obviously the same sign. Two charge q have to compensate for one charge Q that is farther.