Potential of Finite Quadrupole and Zonal Harmonics

In summary, the problem involves finding the potential of an axial quadrupole with point charges q, -2q, and q placed on the z-axis at distances L, 0, and -L from the origin. To find the potential at distances r>>L, a multipole expansion is used and the quadrupole term is found to be non-zero. The potential is then shown to be proportional to one of the zonal harmonics, specifically a Legendre polynomial of degree n=2. The solution involves using the principle of superposition and the binomial expansion.
  • #1
MrCoffee2004
8
0

Homework Statement



a) Find the potential of an axial quadrupole: point charges q, -2q, and q placed on the z-axis at distances L, 0, and -L from the origin.
b) Find the potential only at distances r>>L.
c) Show that this potential is proportional to one of the zonal harmonics.

Homework Equations


The Attempt at a Solution



Ok I am not entirely sure of where to start. I am thinking I need to do an expansion of |r-r'|^(-1) (spherical coordinates) to get to the quadrupole term. Was figuring on having to do a taylor expansion (possibly binomial)? I am not sure however if this is the right approach, and if it is, I am not sure whether to expand with respect to r or r'. It is possible that I have to involve tensors, but I hope not to as we just started talking about them in my Vector and Tensor Analysis class and haven't gone very far with them.

As far as zonal harmonics are concerned, I believe I just have to show it is proportional to one of the legendre polynomials.

I don't want anyone to flat out give me answers, just some guidance :D

Thanks guys
 
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  • #2
a) you can just use the principle of superposition, and you will get a sum.
b) What you have to do is a multipole expansion of the potential. i.e. the multipole expansion of the inverse distance.
You only have to do it till the quadrupole term (this is the term that will be non zero).
 
  • #3
Yea I also started doing a superposition of potentials...

[tex]\varphi[/tex] = [tex]\frac{1}{4\pi\epsilon_{0}}[/tex] [ [tex]\frac{q}{\sqrt{x^{2} + y^{2} + (z+L)^{2}}}[/tex] + [tex]\frac{q}{\sqrt{x^{2} + y^{2} + (z-L)^{2}}}[/tex] + [tex]\frac{-2q}{\sqrt{x^{2} + y^{2} + z^{2}}}[/tex] ]

Noting that r[tex]^{2}[/tex] = [tex]{\sqrt{x^{2} + y^{2} + z^{2}}}[/tex]

[tex]\varphi[/tex] = [tex]\frac{1}{4\pi\epsilon_{0}}[/tex] [ [tex]\frac{q}{\sqrt{r^{2} + 2Lz + L^{2}}}[/tex] + [tex]\frac{q}{\sqrt{r^{2} - 2Lz + L^{2}}}[/tex] + [tex]\frac{-2q}{\sqrt{r^{2}}}[/tex] ]

Little bit of algebra...

[tex]\varphi[/tex] = [tex]\frac{q}{4\pi\epsilon_{0}r}[/tex] [ [tex]\frac{1}{\sqrt{1 + \frac{2Lz + L^{2}}{r^{2}}}}[/tex] + [tex]\frac{1}{\sqrt{1 + \frac{- 2Lz + L^{2}}{r^{2}}}}[/tex] - 2 ]

Not quite sure where to go after that though.
 
  • #4
Forgot to ask, with respect to which variable will I be doing this expansion?..I believe that r and r' are both in the eqn.
 
  • #5
Ok so yea, someone give me a hand? I am basically screwed here.
 
  • #6
Never mind guys, solution came as a superposition of the binomial expansion of the above terms...fell apart and was easy to show to be proportional to legendre polynomial of degree n=2. May post the rest of the work if it will help anyone here.
 

Related to Potential of Finite Quadrupole and Zonal Harmonics

1. What is the significance of finite quadrupole and zonal harmonics in scientific research?

The potential of finite quadrupole and zonal harmonics is an important concept in the study of planetary and atmospheric physics. These harmonics help us understand the gravitational and magnetic fields of planets, which in turn can provide insights into their internal structures and dynamics.

2. How are finite quadrupole and zonal harmonics calculated?

Finite quadrupole and zonal harmonics are calculated through mathematical models that take into account the shape, rotation, and density distribution of a planet. These models use data from satellite missions and ground-based observations to estimate the strength and direction of the gravitational and magnetic fields.

3. Can finite quadrupole and zonal harmonics be used to study other celestial bodies besides planets?

Yes, the concept of finite quadrupole and zonal harmonics can also be applied to other celestial bodies such as moons, asteroids, and even stars. By studying the harmonics of these objects, we can gain a better understanding of their compositions and internal structures.

4. How do finite quadrupole and zonal harmonics affect spacecraft missions?

Knowledge of finite quadrupole and zonal harmonics is crucial for spacecraft missions that involve orbiting or landing on a planet or moon. These harmonics can affect the trajectory and stability of a spacecraft, so accurate calculations are necessary to ensure a successful mission.

5. What advancements have been made in the study of finite quadrupole and zonal harmonics in recent years?

In recent years, advancements in technology and data analysis techniques have allowed for more precise measurements and calculations of finite quadrupole and zonal harmonics. This has led to a better understanding of the internal structures of planets and other celestial bodies, as well as improved accuracy in spacecraft missions.

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