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

In summary, the conversation discusses the calculation of the potential at the axis of a disk with uniform surface charge density. The solution involves using Coulomb's law and the definition of electric potential, but the book introduces a step that is confusing. The book then suggests using a binomial expansion to approximate the electric field at a distance much larger than the disk's radius, which leads to a simplified equation. However, there is a discrepancy in the solution that the book provides, and it may be a typo. The correct equation for the electric field at a large distance should be that of a point charge.
  • #1
zimo
45
0

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...
 
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  • #2
zimo said:
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.
 
  • #3
Doc Al said:
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...
 
  • #4
zimo said:
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:
[tex]\sqrt{z^2 + R^2} = z\sqrt{1 + (R/z)^2}[/tex]

Since R/z << 1, can you see how to use a binomial approximation now?
 
  • #5
Doc Al said:
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.

Doc Al said:
For example:
[tex]\sqrt{z^2 + R^2} = z\sqrt{1 + (R/z)^2}[/tex]

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?
 
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  • #6
zimo said:
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
 
Last edited by a moderator:

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

1. How does Coulomb's law explain the stretching of objects?

Coulomb's law states that the force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This means that as the distance between two charged objects increases, the force between them decreases. When objects are stretched, the distance between the charged particles within them increases, leading to a decrease in the force holding them together.

2. What factors affect the stretching of Coulomb's law?

The stretching of Coulomb's law is affected by the magnitude of the charges on the objects and the distance between them. The larger the charges and the greater the distance, the weaker the force between them and the more likely the objects are to stretch.

3. Is Coulomb's law applicable to all types of objects?

Yes, Coulomb's law is applicable to all types of objects as long as they have a charge. This includes both conductors and insulators.

4. How does the stretching of Coulomb's law relate to everyday objects?

The stretching of Coulomb's law can be observed in everyday objects such as rubber bands, which stretch when pulled due to the repulsive forces between the charged particles within the rubber. Similarly, hair can also become statically charged and repel each other, causing it to stand up or stretch out.

5. Can Coulomb's law be used to explain the stretching of non-charged objects?

No, Coulomb's law only applies to objects with a charge. The stretching of non-charged objects is typically explained by other forces such as tension or gravity.

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