Equilateral triangle charged rods

1. The problem statement, all variables and given/known data



Three rods each of uniform charge Q and length a are formed into an equilateral triangle. Given[Q,a,q] determine:

1. The voltage at the center of the triangle.

2. If a charge -q is placed at the center of the triangle determine potential energy.

3. Determine the initial speed of the -q in order that it exits the system and never returns.

2. Relevant equations





[tex]V = \frac{1}{4 \pi \epsilon_0} \int \frac{dq}{r}[/tex]



[tex]U = Vq[/tex]



[tex]K_0 + U_0 = K + U[/tex]



3. The attempt at a solution



http://i.imgur.com/7IZ3PbP.png



I take 1 half of bottom rod and intend to integrate for V





[tex]dq = Q \frac{dx}{a}[/tex]

[tex]r = \sqrt{y^2 + (\frac{a}{2} - x)^2}[/tex]



using the V formula above this integrates(from 0 to a/2) to





[tex]V = \frac{Q}{4 \pi \epsilon_0}[\ln{\frac{a+\sqrt{\frac{a^2}{4} + y^2}}{y}}][/tex]



since this is 1/6th of the potential and they are additive, multiply 6 for total v at center





is this correct solution? U is simple enough to do, simply multiply by -q. What is the methodology for solving part 3?

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