Tenor strings vs soprano strings?

I am wondering if there is a difference other than length between a tenor string and a soprano string. Being pitch on a string is determined by its diameter tension and length, my initial thought is that because a tenor scale is longer, in order to have the A string play the same A a soprano does, it either needs to be either under more tension or be smaller in diameter. More tension makes the string louder, but it will have less sustain. Same tension, but smaller in diameter will do the opposite...quieter, but more sustain.

First off, are my string physics basically correct?

The reason I ask all this is that I have some tenor strings that I bought, that I am thinking about putting on my soprano, but am wondering if tenor strings are different than the same brand of soprano strings.

Yes - there is a difference in string diameters. Here are specs for D'Addario Ukelele string sets. The Worth site provides string diameters but not tensions. Just Strings shows diameter and tension of Aquila string sets.