Here is my solution. It seems that this problem was more conceptual than hard-core computation.
The Poisson equation for the potential is:
This is a linear partial differential equation, and the uniqueness and existence theorem states that if this equation has a solution for certain ...
This solution includes calculations for electric force and fields.
Working with electric forces, field and potentials.
Two positive charges, charge q0 are fixed on the x axis at positions +a and -a.
a) What is the Electric Field anywhere in the (x,y) plane, at position (x,y)? Be sure your answer clearly denotes the vector nature of the electric field.
b) What is the electric potential at position (x,y)?
c) A third charge, q3 , (mass m3 ) is at position (0, 0 y ). What is the force on this charge?
d) What is the minimum speed of the third particle at position (0, 0 y ) in order for it to
(i) reach the origin , (0,0)
(ii) move off to infinity (plus or minus).
Be sure to consider both signs of 3 q to answer these.
The answers may not algebraically reduce to 'nice' solutions in general. Be sure to present in terms of defined variables.