Approximation of linear quadrupole as a dipole in radiation zone

[dorist84]

Hello,

This question is more conceptional - I think I can do the algebra (mostly approximations) in this problem ok.

Homework Statement

I am wondering why it is for a radiation zone, that a linear electric quadrupole can be approximated as a dipole. I am wondering if this is just a coincidence or if this is a specific case due to an intrinsic geometry or a characteristic about radiation zones that allows us to do this.

I doubt it is coincidental - if it were, I would guess that higher order dipoles (meaning bigger than quadrupoles) can be approximated as such then - that doesn't make sense to me.

There must be something "special" about radiation fields that allows us to say this: does this have to do with the energy density then? Maybe the magnitude of the Poynting vector? Perhaps the amplitude of the wave oscillation in the B-field in the radiation zone...(But what would dipoles/quadrupoles have to do with this then?)

Anything clarification would be greatly appreciated. Thanks so much! Take care.

--Doris

Homework Equations

The Attempt at a Solution

 

[BigJDubs]

The reason why a linear electric quadrupole can be approximated as a dipole in a radiation zone is due to the fact that the fields created by the quadrupole and the dipole are both produced by the same underlying source. The underlying source in this case is the oscillatory behavior of the electric and magnetic fields caused by the radiation. The magnitude of the Poynting vector, which is related to the amplitude of the wave oscillation in the B-field, will be the same for both the quadrupole and dipole. Therefore, the fields generated by both sources will be the same, which allows us to approximate a quadrupole as a dipole in a radiation zone. Higher order dipoles will not be able to be approximated as such because the fields produced by higher order dipoles are not necessarily related to the oscillatory behavior of the electric and magnetic fields as the quadrupole and dipole are. This is why a higher order dipole cannot be approximated as a dipole in a radiation zone.