Electrostatic Force
Four particles form a square. The charges are $\displaystyle q_1 = q_4 = Q$ and $\displaystyle q_2 = q_3 = q$.
Just for your info, the particles are arranged as such:
1 2
3 4
(a)What is $\displaystyle \frac Qq $ if the net electrostatic force on particles 1 and 4 is zero?
Here's my solution for part a:
(My notation for Force for particles is this, say I want to say the force of particle 2 on 1, I write $\displaystyle F_{21}$)
$\displaystyle \sum F_1 = F_{21} + F_{31} + F_{41}$
This says that the net force on 1 is the sum of the forces on 1 from 4, 3, and 2.
$\displaystyle \sum F_1 = k\frac{qQ}{a^2} + k\frac{qQ}{a^2} + k\frac{Q^2}{(\sqrt{2}a)^2}$
$\displaystyle = 2k\frac{qQ}{a^2} + k\frac{Q^2}{2a^2}$
$\displaystyle 2k\frac{2qQ + Q^2}{2a^2} = k\frac{2qQ + Q^2}{a^2}$
Turns out I get the same thing for $\displaystyle \sum F_4$ because the net force for Q is 0. So, now I get the net force on Q:
$\displaystyle \sum F_Q = 2k\frac{2qQ + Q^2}{a^2} = 0$
$\displaystyle \frac{2k}{a^2}(2qQ + Q^2) = 0$
Since k and a are obviously not 0, then $\displaystyle (2qQ + Q^2)$ must be 0, so:
$\displaystyle 2qQ + Q^2 = 0$
$\displaystyle Q^2 = 2qQ$
$\displaystyle Q = 2q$
$\displaystyle \frac Qq = 2.0$
Is that right? If so, here's part b.
(b) Is there any value of q that makes the net electrostatic force on each of the four particles 0?
If my answers aren't right, could you please correct them and explain so I can better understand how to do other problems of this nature. Thanks.
