3D Vector Field of a given function on a sphere

Questions
,
1

I want to plot vector fields on a sphere of a given function.
Let’s say that I have this function:
field_eq.png

Where a, b are radii of the spheres (known parameters), U is the applied voltage on the outer sphere (also, known). Obviously, the equation is in spherical coordinate system, with r the radial distance.
It represent electric field, and how can I plot it on a sphere? I’ve tried with VectorPlot3D, but the results are not really accurate:

topka.gif
topka.gif1560×1560 314 KB

Any help, tips and suggestions how can I plot the vectors which “follow” this equation? ( their intensity has to change as the applied voltage changes, as well as the radial distance of the sphere center )

Thanks!

2

Hello,
I am not sure if I understand your question, so please excuse my answer if I have misunderstood.
How about something like this:

With[{points = SpherePoints[24], length = 10},
Manipulate[
Graphics3D[{{Yellow, Opacity[0.5], Sphere[{0, 0, 0}, 10 r]}, 
Arrow[Tube[{10 r #, 10 r # + # length/r^3}]] & /@ points}, 
PlotRange -> 40 {{-1, 1}, {-1, 1}, {-1, 1}}],
{{r, 1}, .5, 10}
]
]

I made the visualization sphere artificially large here, but that helps with the visualization I think.