Asymptotic behavior quadrupole potential

In summary, the asymptotic behavior of a quadrupole consisting of a -2 charge at the origin and +1 charges at z = +/- 1 can be described as 1/r, but this is not entirely accurate due to the cancellation of charges at large distances. To incorporate this, one can Taylor expand the expression for the total potential about the point 1/r=0.
  • #1
NanakiXIII
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Homework Statement



Find the asymptotic behavior of a quadrupole consisting of a -2 charge at the origin and +1 charges at z = +/- 1.

Homework Equations



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The Attempt at a Solution



You can construct the Coulomb potentials for these three charges and show that for large r, they all go as 1/r. The problem is that if you add them together, for large r, the -2 cancels the 2*1, so for large r you actually get 0/r. This makes the potential go to zero more quickly than 1/r, and 1/r doesn't seem like a fitting description of the asymptotic behavior. How do I go about incorporating this?
 
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  • #2
Taylor expand your expression for the total potential about the point 1/r=0
 
  • #3
Yup, that worked. Thanks.
 

1. What is the definition of asymptotic behavior in the context of quadrupole potential?

Asymptotic behavior refers to the behavior of the quadrupole potential as the distance from the source of the potential approaches infinity. In other words, it describes how the potential changes as the distance from the source increases indefinitely.

2. How is asymptotic behavior of quadrupole potential related to the inverse square law?

The asymptotic behavior of quadrupole potential follows the inverse square law, meaning that as the distance from the source increases, the potential decreases according to an inverse square relationship. This is because the potential is spread out over a larger area as the distance increases.

3. Can asymptotic behavior of quadrupole potential be used to determine the strength of the source?

Yes, the asymptotic behavior of quadrupole potential can be used to determine the strength of the source. By studying the rate at which the potential decreases as the distance increases, the strength of the source can be calculated using mathematical equations.

4. How does the shape of the source affect the asymptotic behavior of quadrupole potential?

The shape of the source does not affect the asymptotic behavior of quadrupole potential. As long as the source is symmetrical and has a quadrupole moment, the potential will follow the same inverse square law behavior regardless of its shape.

5. What is the significance of studying the asymptotic behavior of quadrupole potential?

Studying the asymptotic behavior of quadrupole potential is important in understanding the behavior of electric and magnetic fields in a wide range of physical systems. It can also provide insights into the properties and interactions of particles such as protons and neutrons, which have quadrupole moments.

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