Does this answer look right? (Infinitely long conductor pipe)

In summary, the conversation discusses the voltage distribution between the inner and outer surface of an infinitely long conducting pipe, which is governed by Laplace's equation in polar coordinates with given boundary conditions. The solution for V is V = (Vc log(r) - log(R2))/(log(R1) - log(R2)), or V(r, θ) = Vc * [log(r) - log(R2)] / [log(R1) - log(R2)].
  • #1
Jamin2112
986
12

Homework Statement



Fig. 1 shows the cross-section of an infinitely long conducting pipe. The inner radius of the pipe is r = R1, and the outer radius is r = R2. Suppose the inner surface has a constant voltage V = Vc > 0, and the outer surface has a constant voltage V = 0. The voltage distribution between the inner and outer surface (blue region) is governed by the Laplace's equation ∇2V=0 expressed in polar coordinates. Solve for V with the given boundary conditions.

[Note: Fig. 1 just looks like a blue annulus centered at the origin in the plane.]

Homework Equations



Don't worry about them for now.

The Attempt at a Solution



Just tell me whether my solution V = (Vc log(r) - log(R2))/(log(R1) - log(R2)) intuitively seems like it's correct. I don't want you to carry out the process of deriving this ... unless you want to.
 
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  • #2
I.e. the voltage at a distance r from the center of the piper is proportional to log(r)
 
  • #3
EDIT:

Actually I got

V(r,θ) = Vc * [log(r) - log(R2)] / [log(R1) - log(R2)].​

Does that seem right?
 

1. What is an infinitely long conductor pipe?

An infinitely long conductor pipe is a hypothetical object used in physics and engineering to simplify calculations and models. It is a long, straight pipe with a uniform distribution of charge or current along its length. This means that the amount of charge or current is the same at every point on the pipe.

2. How is the concept of an infinitely long conductor pipe useful?

The concept of an infinitely long conductor pipe is useful because it allows us to simplify complex calculations involving electric or magnetic fields. By assuming that the pipe extends infinitely in both directions, we can ignore edge effects and focus on the behavior of the field at a specific point.

3. What is the relationship between an infinitely long conductor pipe and Gauss's Law?

An infinitely long conductor pipe is often used in applications of Gauss's Law, which states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space. The pipe's uniform charge distribution allows us to easily calculate the electric field at any point using Gauss's Law.

4. Can an infinitely long conductor pipe exist in reality?

No, an infinitely long conductor pipe is a theoretical construct and cannot exist in reality. In reality, all objects have a finite length and cannot extend infinitely in both directions. However, the concept of an infinitely long conductor pipe is still useful for simplifying calculations and understanding the behavior of electric and magnetic fields.

5. How does the behavior of an infinitely long conductor pipe differ from that of a finite conductor pipe?

An infinitely long conductor pipe and a finite conductor pipe behave differently in terms of their electric and magnetic fields. In an infinitely long pipe, the electric field is constant along its length, while in a finite pipe, the field decreases as you move away from the center. Additionally, the magnetic field of an infinitely long pipe is uniform around the pipe, while a finite pipe will have varying magnetic field strengths at different points along its length.

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