In this link http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S4 , Feynman first computes the gravitational field generated by an infinite plane of constant density, and then he computes the gravitational field generated by a hollow sphere (also with constant density). Actually, in the second case he doesn't exactly compute the field but the potential, which works out easier since it's a scalar and not a vector. I then thought of trying to do the same thing for the plane case, and I worked out the math, and I don't know if I'm doing something wrong, but it doesn't work.
Here's what I'm doing: let's use Feynman's notation in figure 13-5. As he computes it, the mass of the infinitesimal ring is $dm = \mu 2\pi \rho d\rho$. As he mentions, we have $\rho d\rho = r dr$ (from the Pythagorean theorem). Well, then, to me, it only stands to reason that the potential due to this infinitesimal ring is $$U = - GMdm/r = -GM(\mu 2\pi r dr)/r = -GM\mu 2\pi dr$$ (here I'm calling $M$ the mass of the point above the plane, which he ignores in his calculations but it makes no difference)
But now the problem is clear: this $U$ is constant, so if we integrate $r$ from $a$ to $\infty$ we will just get infinity. What am I doing wrong? I'm pretty confused.
