Vector Spherical harmonics/spherical coordinates question

[Faust1]

Arfken and Weber lists some of the Vector spherical harmonics in spherical coordinates, and I'm puzzled that one has no radial component. Specifically, $Y(j =1, l=1, m = 0) = i \sqrt[3/(8 \Pi)] sin(\theta) \^{phi} $

To Cartesian components of the vector, it seems you need an r component of the vector in spherical coordinates. Since Y110 only has a phi component, does this mean that the Y110 spherical harmonic is a vector of length zero in cartesian coordinates? If so, why is there even a magnitude in the phi component of the vector? And if so, is Y110 only useful if combined with other vector spherical harmonics?

Thanks in advance.

 

[LightHero]

The answer to your question is yes, Y110 in spherical coordinates is a vector of length zero in Cartesian coordinates. This is because the radial component of Y110 is zero, and the other two components are related to each other by the spherical coordinates system. Therefore, Y110 does not have a meaningful magnitude in spherical coordinates, but it is still useful as it helps us to describe the direction of a vector in spherical coordinates. For example, if you have a vector with components (r, θ, φ), then Y110 will tell you the direction of the vector in terms of θ and φ.