Derivation of electric field for a dipole.

[Scintillation]

Homework Statement

A test charge P is separated by a distance "Z" from the midpoint of a dipole. The distance between the two particles in the dipole is d. Find an equation for the electric field between the dipole and the test charge.

The Attempt at a Solution

Since I did not want to spend a lot of time in Latex, I did a quick sketch of my work in Paint (thus, the messiness).

I am following Halliday/Resnick 8th edition. The book does not explain most of this problem, so I redid this based on class notes.

http://i43.tinypic.com/2jbf7.jpg

I'm having an issue with some of the steps. I understand that there is a superposition of electric fields from the negative and positive charges, and used the basic equation (kq/r^2) to solve for the separate electric fields.

Unfortunately, the book calls skips some of the steps, calling it "some algebra," and my instructor went through it very quickly.

I highlighted the problem areas:
1. I don't understand how (z^2 + (d/2)^2)= z^2(1+(d/2z).
How exactly does that work? That is very confusing.

2. Supposedly, I can expand this part using the binomial theorem.
But the binomial theorem is (1+x)^n= 1 + (nx) + (n(n-1)^2)/2!

Given that n=-2, I don't know why I can skip the next step.

After these two problem areas, I understood the rest of it, and am able to solve (as shown). Yet, these two areas that I didn't understand are pretty important, and it would be terrible to simply memorize them.

 

[SammyS]

...
Unfortunately, the book calls skips some of the steps, calling it "some algebra," and my instructor went through it very quickly.

I highlighted the problem areas:
1. I don't understand how (z^2 + (d/2)^2)= z^2(1+(d/2z).
How exactly does that work? That is very confusing.
...

The notes don't say (z² + (d/2)²)= z²(1+(d/2z) .

They say (z² + (d/2)²)= z²(1+(d/2z)²) .

That's just algebra, factoring out z² .

Do you need that explained further?