Gap between two displaced spheres

[Opus_723]

This was out of a physics book, but since it's just geometry, I thought this would be the best forum to ask about it.

I was reading through a derivation of the far electric field of a uniformly polarized sphere, and the author used a trick where he modeled the sphere as two displaced spheres of uniform charge density. As in, spheres that were originally overlapping, but then one is shifted up by a small distance and the other shifted down the same. The idea was that geometrically, the gap between the spheres varies exactly as cosθ, which is what was needed for the problem.

He didn't prove that the gap varied as cosθ. It's pretty intuitive just from looking at the figure (sort of like a venn diagram). But I figured I would prove it to myself just as an exercise. Anyway, it turned out to be harder than I thought, and I can't get a simple cosine function to fall out. I tried writing out the equations of two circles centered at small displacements from the origin and then using the distance formula for each, trying to find the difference in distances from the origin at every point, but I wasn't able to simplify the square roots far enough to get anything clear. I feel like I'm probably overlooking something simple.

Could anyone direct me to a proof of this property?

 

[tiny-tim]

Hi Opus_723!

If you have two identical circles of radius r, and one is shifted a small distance h upward,

then obviously the gap measured vertically at any point is h,

but you want the gap measured radially, which will be at an angle θ, and so is hsinθ