Potential of an infinite rod using Green's function

In summary, the conversation discusses a homework problem in which an infinite line of constant charge density is considered. The individual suggests using a formula given in a link to find the equipotentials, but realizes that the potential is infinite everywhere. Another individual points out that this formula only works for localized sources, while the source in this problem extends to infinity. The conversation concludes with the understanding that the problem cannot be solved using the given formula as the charge density does not satisfy certain properties.
  • #1
fluidistic
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Homework Statement


Same problem as in https://www.physicsforums.com/showthread.php?t=589704 but instead of a spherical shape, consider an infinite line of constant charge density [itex]\lambda _0[/itex].


Homework Equations


Given in the link.


The Attempt at a Solution


I assume Phi will be the same along any parallel line to the charge distribution. So I can calculate [itex]\Phi (y)[/itex] in an x-y plane where x is the line direction and y=0 is where the line is.
I get that [itex]|\vec x - \vec x'|=\sqrt {x^2+y^2}[/itex] so that [itex]\Phi (y)=\int _{-\infty }^{\infty } \frac{\lambda _0 dx}{\sqrt {y^2+x^2}}=\lambda _0 \ln ( \sqrt {y^2+x^2 } ) \big | _{-\infty}^{\infty}[/itex] but this diverge.
By intuition I know that the equipotentials must be parallel to the charge distribution but here I get that the potential is infinite everywhere.
I don't know what's going on.
 
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  • #2
Again this time you'll need to subtract an infinite constant to make your potential finite and meaningful. The green's function you used assumes the sources are localized, i.e., zero boundary condition at infinity, whiles your source extend to infinity, therefore it won't work.
 
  • #3
sunjin09 said:
Again this time you'll need to subtract an infinite constant to make your potential finite and meaningful. The green's function you used assumes the sources are localized, i.e., zero boundary condition at infinity, whiles your source extend to infinity, therefore it won't work.
Hmm so I cannot solve the problem via the given formula?
 
  • #4
My professor said today that the problem cannot be solved via the integral formula given because rho does not satisfy some properties. The problem was ill posed. :smile:
 

FAQ: Potential of an infinite rod using Green's function

What is Green's function and how is it related to the potential of an infinite rod?

Green's function is a mathematical tool used in solving differential equations. It represents the response of a system to a point source. In the context of the potential of an infinite rod, Green's function is used to calculate the potential at a point due to a point source located anywhere along the rod.

How is the potential of an infinite rod calculated using Green's function?

The potential at a point due to a point source located at a distance r from the point can be calculated using the formula: V(r) = G(r) * q, where G(r) is the Green's function and q is the strength of the point source. The potential at a point due to multiple point sources can be calculated by summing the potentials due to each individual source.

What are the assumptions made in using Green's function to calculate the potential of an infinite rod?

The main assumptions are that the rod is infinitely long, has a uniform charge distribution, and is infinitely thin. Additionally, the potential is only valid outside the rod, as the potential inside the rod is infinite due to the infinite charge density.

How does the potential of an infinite rod using Green's function differ from other methods of calculation?

The potential of an infinite rod can also be calculated using other methods, such as the method of images or using the electric field due to the rod. However, Green's function provides a more general and elegant solution that can be applied to a wide range of problems involving point sources.

Can Green's function be used to calculate the potential of a finite rod?

Yes, Green's function can also be used to calculate the potential of a finite rod. However, the formula for the potential will be different and will depend on the length and size of the rod. In this case, the potential will also be valid both inside and outside the rod.

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