Proof that disk of charge = point charge when very far?

[Brennen berkley]

Homework Statement

Take the expression 21.11 (pictured below, specifically the bottom one) for the electric field above the center of a uniformly charged disk with radius R and surface charge density σ, and show that when one is very far from the disk, the field decreases with the same square of the distance as it would for a point charge, E ~ q/4πεx², where q is the total charge on the disk.

Homework Equations

The Attempt at a Solution

I solved one like this pretty easily where you prove that a long wire acts as an infinite wire, but I've been looking at this one for about an hour and I'm stumped. I know I need to get x² on the bottom of the equation without being inside a square root, but I don't know how. The only approximation I can think of is that when R is much bigger than x, √(x² + R² + 1) goes to 1, but then I don't have an x anymore.

 

[TSny]

Letting R become much bigger than x is the opposite of the approximation you want here.

Binomial expansion might help: https://en.wikipedia.org/wiki/Binomial_approximation

 

[haruspex]

I think you mean x much larger than R.
Do you know the binomial expansion of (1+t)^a for small t?

Edit: must learn to post more quickly.

 

[Brennen berkley]

Yes, I meant if x is much larger than R then √(R²/x² + 1) goes to 0 (hopefully I said it right that time). I haven't done binomial expansions, for a while so I'll review those, thanks.

 

[Brennen berkley]

Ok I'm still stuck. I used σ = q/πR² and did the binomial expansion and got this:

EDIT: that should be R²/2x²

 

[TSny]

Note that $\frac{1}{\sqrt{1+t^2}} = (1+t^2)^{-1/2}$

 

[Brennen berkley]

I see how to do it know, it's pretty simple, thanks.