How did it get to that? Stretching Coulomb's law

[zimo]

Homework Statement

The charge is distributed with uniform
surface density σ on the disk of radius R. Find the potential
at the axis of the disk.

Homework Equations

Coulomb's law and the definition of a electric potential at point x

The Attempt at a Solution

I have a solution in front of me but can't understand some step inside it:

The potential can be defined now

phi(x)= (1/4pi*epsilon0)Integral[(sigma(x')/|x-x'|)dS'] and the solution for the integral from 0 to R is:

(sigma/2*epsilon0)(sqrt(x^2+R^2)-z)

Now, the electric field at this point is:

E(z)=(sigma/2*epsilon0)(1-(z/sqrt(z^2+R^2))

I can clearly follow until now, but then the book says that for z>>R we get

E=Q/(4pi*epsilon0*z)

where Q is the total charge of the disc.

How can it be proportional to 1/z?? when I take z>>R - I get E(z)=0...

 

[Doc Al]

I can clearly follow until now, but then the book says that for z>>R we get

E=Q/(4pi*epsilon0*z)

Are you sure that z isn't z²?

How can it be proportional to 1/z?? when I take z>>R - I get E(z)=0...

The point is to see how the field drops off. Hint: Do a binomial expansion.

 

[zimo]

Are you sure that z isn't z²?


The point is to see how the field drops off. Hint: Do a binomial expansion.

1. Yeah, I'm sure. I double checked it.

2. what kind of a binomial expansion can I possibly make here? please direct me some more...

 

[Doc Al]

1. Yeah, I'm sure. I double checked it.

I'd say that was wrong. What book are you using?

2. what kind of a binomial expansion can I possibly make here? please direct me some more...

For example:
$$\sqrt{z^2 + R^2} = z\sqrt{1 + (R/z)^2}$$

Since R/z << 1, can you see how to use a binomial approximation now?

 

[zimo]

I'd say that was wrong. What book are you using?

http://www.ph.biu.ac.il/data/teach/l/86-234/electrodynamics11.pdf" , third page.

For example:
$$\sqrt{z^2 + R^2} = z\sqrt{1 + (R/z)^2}$$

Since R/z << 1, can you see how to use a binomial approximation now?

You mean the approximation is ~= 1+z^2/2*R^2?

 

[Doc Al]

http://www.ph.biu.ac.il/data/teach/l/86-234/electrodynamics11.pdf" , third page.

OK, but I think it's a typo. As you get far enough away, its field should be that of a point charge.

You mean the approximation is ~= 1+z^2/2*R^2?

Something like that: √(1 + R²/z²) ≈ 1 + ½R²/z²