Multipole Expansion of a Thin rod

In summary, the problem involves calculating the potential of a thin rod with uniform charge density at a point away from the rod. The solution involves using cartesian coordinates and treating theta as a constant. However, there may be an error in the question and further algebraic checks are needed.
  • #1
sitkican
2
0

Homework Statement


Consider a very thin rod lying on the z axis from z = −L/2 to z = L/2. It carries a uniform charge density λ. Show that away from the rod, at the point r (r >>L), the potential can be written as V (r, θ) = (2Lλ/4πε0)(1/L)[ 1 + 1/3(L/2r)2P2(cos θ) + 1/3(L/2r)4 P4(cos θ) + · · ·# (1)

Homework Equations


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The Attempt at a Solution


At the beginning, I tried to solve this question using cylindrical coordinates as there is some theta dependence in Legendre polynomial but it did not work.Then,I took the volume integral using cartesian coordinates and treating theta as constant.By doing that I could just take the Legendre polynomials out of the integration and calculate the integral with respect to z. I calculated the integral and put values of n up to n=4 and get the exact result in the square brackets but I have λL/4πε0r outside while I have to have 2λL/4πε0L.

What should I do know? Is my assumption,the theta is constant, true?

Thanks for your help...
 
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  • #2
Check your algebra.
 
  • #3
vela said:
Check your algebra.
thank you for your comment. the question is wrong, I informed today.
 

1. What is the multipole expansion of a thin rod?

The multipole expansion of a thin rod is a mathematical technique used to describe the electric field produced by a thin, elongated object with a uniform charge distribution. It involves breaking down the electric field into a series of terms, each representing a specific type of charge distribution.

2. How is the multipole expansion of a thin rod calculated?

The multipole expansion of a thin rod is typically calculated using the method of moments, which involves integrating the charge distribution of the rod over its length. This results in a series of terms, each representing a specific type of charge distribution, such as a dipole, quadrupole, etc.

3. What is the physical significance of the multipole expansion of a thin rod?

The multipole expansion of a thin rod allows us to approximate the electric field produced by the rod at a distance, without having to consider the details of its charge distribution. This is useful in situations where the rod is too complex to analyze using other methods, or when we only need a rough estimate of the electric field.

4. How does the multipole expansion of a thin rod relate to other multipole expansions?

The multipole expansion of a thin rod is just one specific case of a more general multipole expansion, which can be used to describe the electric field produced by any object with a non-uniform charge distribution. However, the thin rod is unique in that it has a simple, symmetric charge distribution that allows for a simpler and more intuitive calculation of the multipole terms.

5. What are some real-world applications of the multipole expansion of a thin rod?

The multipole expansion of a thin rod has many practical applications, such as in antenna design and analysis, where the electric field produced by a thin, elongated antenna can be approximated using the multipole expansion. It is also used in plasma physics, where it can be used to describe the electric field produced by a thin, charged plasma sheet. Additionally, the multipole expansion is used in astrophysics to analyze the electric fields produced by rotating bodies, such as planets and stars.

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