Find locus of points in the plane of the charges where the field is zero?

Two electric charges q1=q and q2= -2q are placed at a distance L = 6a apart. Find the locus of points in the plane of the charges where the field potential is zero.|||The electric potential at a distance (r) from a point charge of charge q is given by:





f = q/(4*pi*e*r)





where e is the permittivity of the medium.





The potential due to a system of point charges is given by the sum of the potentials of the individual charges.





In this case, let the charges q and -2q be located on the x axis, at points x = -3a and x = +3a, respectively. The potential due to the two charges is then given by:





F = q/(4*pi*e*sqrt((x + 3a)^2 + y^2)) - 2*q/4*pi*e*sqrt((x - 3a)^2 + y^2))





We want to know the locus of x,y points that result in F = 0





0 = q/(4*pi*e*sqrt((x + 3a)^2 + y^2)) - 2*q/4*pi*e*sqrt((x - 3a)^2 + y^2))





0 = 1/sqrt((x + 3a)^2 + y^2) - 2/sqrt((x - 3a)^2 + y^2))





2/sqrt((x - 3a)^2 + y^2)) = 1/sqrt((x + 3a)^2 + y^2)





2*sqrt((x + 3a)^2 + y^2) = sqrt((x - 3a)^2 + y^2))





4*((x + 3a)^2 + y^2) = (x - 3a)^2 + y^2)





3y^2 = (x-3a)^2 - 4*(x+3a)^2





y^2 = -x^2 -10x*a -9a^2





y = +sqrt(-(x+9a)(x+a)) and y = -sqrt(-(x+9a)(x+a))