Potential of a Quadrupole (Far Away)

[ToothandnaiL]

So we're currently covering multipole expansion techniques in my EDM class. The multipole expansion is a summation of integrals each treating a different configuration of discrete charges (monopole, dipole... and so on). The term in the summation that corresponds to the physical configuration that your dealing with, whether it be a monopole, dipole, or whatever, is the dominant term. My first question is: Does that mean it is safe to ignore the other terms in the summation and treat, say, a quadrupole with only the quadrupole term? Secondly, there is a simple formula to express the potential of a discrete charge configuration at a large distance where the configuration appears as a single point charge. The problem I'm looking at has a charge configuration like this: a +3q charge at point z=a, a +q charge at z=-a, a -2q charge at y=a and y=-a, this forms a quadrupole and the problem wants an expression for potential that is valid at large distances from the quadrupole. It uses the word 'simple' to describe the expression for potential so I imagine it's not the multipole expansion formula. Would I use the formula for large distances to treat this problem? The only problem I see with that would be the fact that the net charge would=0 and then the potential would= 0 as well. How could that be the case for this quadrupole?

 

[Muphrid]

A multipole expansion is just a expansion of the potential in terms of powers of $1/|r-r'|$. If you can verify that a certain collection of charges has neither a net charge nor a dipole moment and that the quadrupole moment is nonzero, it is guaranteed that the quadrupole contribution is dominant over all others far away from the charge distribution.

 

[mfb]

Does that mean it is safe to ignore the other terms in the summation and treat, say, a quadrupole with only the quadrupole term?

It depends on the required precision. If you want to know the orbital period of moon to calculate the days of full/new moon, it is fine to treat Earth as a monopole. If you want to know its position with an accurary of centimeters, you have to add more terms.

a +3q charge at point z=a, a +q charge at z=-a, a -2q charge at y=a and y=-a, this forms a quadrupole and the problem wants an expression for potential that is valid at large distances from the quadrupole.

What about the dipole moment?

I think "simple" means the highest order is sufficient.

 

[ToothandnaiL]

OK, thank you for the replies they give some new ways to think about the problem.

 

[PJC01]

I would advise you to carefully consider the assumptions and limitations of the multipole expansion technique. While it may be appropriate to ignore certain terms in the summation for a specific physical configuration, it is important to understand the consequences of doing so. In the case of a quadrupole, ignoring terms may result in an inaccurate representation of the system and could lead to errors in your calculations.

Regarding the formula for the potential of a discrete charge configuration at a large distance, it is important to note that this formula is only valid for a single point charge. In the case of a quadrupole, the net charge may be zero, but the distribution of charges is still important in determining the potential at a large distance. Therefore, using the formula for a single point charge may not accurately reflect the potential of the quadrupole.

In this situation, I would recommend using a more rigorous approach, such as the method of images or solving the problem using the Laplace equation. These methods take into account the full charge distribution and can provide a more accurate representation of the potential at a large distance from the quadrupole. It is important to carefully consider the assumptions and limitations of any formula or technique used in scientific calculations to ensure accurate results.