I'm having a lot of trouble solving for part b as I am unable to correctly apply the binomial theorem to this approximation. The problem is shown below:
Three point charges are distributed: a positive charge +2Q in the center, and a pair of negative charges -Q, a distance a to its left and right.
You want to find the electric field E at point P, a distance r to the right of the positive charge.
a) Write an exact expression for the x-component of E at point P (in terms of Q,r,a, etc.
b) Using the binomial theorem, write a simpler approximation to the expression you gave in part b, which is valid when r>>a
Binomial theorem: (1+x)^n=1+nx+n(n-1)x^2/2!+...
The Attempt at a Solution
a) I added the vector sum of all forces and got 1/(4pi epsilon knot)(-Q/(r+a)^2+(2Q)/r^2+(Q)/(r-a)^2
b) The professor gave out the answer for this part which is 6kQa^2/r^4, but we need to show how we got this. I go as far as KQ(2-(a-2a/r)-(1+2a/r))r^2, but I don't know how to proceed further. I wish I knew how to type out the work more neatly so you can see clearly. I would greatly appreciate any feedback.