Green's function for infinitely long cylinder

This satisfies the boundary conditions that we need (R(a) = V0 and R(∞) = 0).For the second equation, we can use the method of images to find the appropriate solution. This means that we introduce a "mirror" cylinder at r = -a, which has the same potential as the original cylinder but with opposite sign. This gives us the following solution for F:F(θ,z) = -V0/2π - V0/2πln(r/a) for r < aF(θ,z) = -V0/2π + V0/2πln(r/a) for r > aNote that this satisfies the boundary conditions (F(a)
  • #1
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Homework Statement



Find the Green's function for the Dirichlet boundary conditions for the interior of an infinite cylinder of radius a.

Homework Equations



[tex] \nabla^2 G(x,x') = -4 \pi \delta(x-x') [/tex]

and in general, Green's functions are of the form

[tex] G(x,x') = \frac{1}{|x-x'|}+F(x,x') [/tex]

where F(x,x') satisfies the Laplace equation.

The Attempt at a Solution



All I know, really, is that we have the two relations above. I guess I feel like I know why we have a solution with the Green's functions - as in, I know what to do with them once I've defined them - but I'm at a loss as to how to define them.

Do I just take F(x,x') to be the general solution to the Laplace equation in polar coordinates and then add that to the 1/|x-x'| term?

And I'm not sure quite what to make the boundary conditions. They're Dirichlet, so we need to demand that G(x,x') = 0 for x' on the cylinder. We also need to demand that the potential doesn't blow up inside the cylinder. But this problem didn't specify what sort of surface the cylinder is (we can probably assume it's some sort of conductor, but there are no specifics) and there's no mention of any charge density anywhere. So, yeah, I'm a bit confused... can anyone help?
 
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  • #2


Thank you for your post. Finding the Green's function for this problem can be a bit tricky, but I'll try my best to explain the steps.

First, let's define the problem a bit more clearly. We have an infinite cylinder of radius a, and we are looking for the Green's function for the interior of this cylinder, subject to Dirichlet boundary conditions. This means that the potential at the surface of the cylinder is fixed to some value (let's call it V0), and the potential inside the cylinder must satisfy Laplace's equation.

Now, let's use cylindrical coordinates (r, θ, z) to simplify the problem. We can write the Laplace equation in these coordinates as follows:

∂^2G/∂r^2 + (1/r)∂G/∂r + (1/r^2)∂^2G/∂θ^2 + ∂^2G/∂z^2 = -4πδ(r-a)

Note that the delta function only appears in the radial coordinate, since we are only concerned with the potential inside the cylinder.

Next, we can use the method of separation of variables to solve for G. Let's assume that G can be written as a product of two functions, one that only depends on r and one that only depends on θ and z. This gives us the following equations:

(1/r)(∂/∂r)[r(∂G/∂r)] + (1/r^2)(∂^2G/∂θ^2 + ∂^2G/∂z^2) = -4πδ(r-a)

∂^2R/∂r^2 + (1/r)∂R/∂r = -4πδ(r-a)/r

∂^2F/∂θ^2 + ∂^2F/∂z^2 = 0

Note that we have separated out the delta function and the Laplacian in cylindrical coordinates. Now, we can solve these equations individually.

For the first equation, we can use the fact that the delta function only appears in the radial coordinate to solve for R. We can write R as a piecewise function:

R(r) = A for r < a

R(r) = B for r > a

where A and B are constants to
 

Related to Green's function for infinitely long cylinder

1. What is the purpose of using Green's function for infinitely long cylinder?

The purpose of using Green's function for infinitely long cylinder is to solve boundary value problems in electromagnetics, specifically for a conducting cylinder with an infinite length. It is a mathematical tool that helps to find the electric and magnetic fields in and around the cylinder.

2. How is Green's function for infinitely long cylinder different from other Green's functions?

Green's function for infinitely long cylinder is unique because it takes into account the boundary conditions of an infinitely long cylinder, which is different from other Green's functions that are used for different geometries. It also has a singularity at the origin due to the cylindrical symmetry of the problem.

3. What are the applications of Green's function for infinitely long cylinder?

Green's function for infinitely long cylinder has various applications in electromagnetics, such as in the analysis of antennas, waveguides, and transmission lines. It is also used in the design and optimization of microwave devices and circuits.

4. How is Green's function for infinitely long cylinder calculated?

Green's function for infinitely long cylinder is calculated using the method of images, where the electric and magnetic fields are expressed as a sum of fields from a point source at the origin and an infinite series of images. This allows for the solution of the boundary value problem in terms of known solutions for simpler geometries.

5. What are the limitations of using Green's function for infinitely long cylinder?

One limitation of using Green's function for infinitely long cylinder is that it can only be applied to problems with cylindrical symmetry. It also assumes that the cylinder is a perfect conductor and does not take into account any losses or imperfections. Additionally, the mathematical calculations involved can become complex for more complicated geometries, making it difficult to obtain an analytical solution.

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