To solve the gold nugget net charge problem, first determine the mass of a gold atom and calculate the total number of gold atoms in a 1.0-gram nugget. Next, calculate the total number of electrons by multiplying the number of gold atoms by the atomic number of gold. After removing 1.0% of the electrons, the net charge can be found by considering the remaining electrons and the protons present. The discussion also briefly touches on calculating the electric field inside and outside a uniformly polarized cylinder, indicating a mathematical relationship for the external field.