We will calculate the first three multipole moments of a deformed charged sphere. Such a charge distribution is often used to model nuclei and understand their stability. The rotational symmetry of the problem will ease our computations significantly.
In rotational symmetry, the multipole moments of a charge distribution are described by just one component per order l. This is a drastic difference to the usual 2l + 1 independent components. Get to know this simplifications with two examples and derive it for any charge distribution with rotational symmetry!
The quadrupole moments of a homogeneously charged ellipsoid are calculated. The result will help us to understand how and why the quadrupole moments of an ellipsoidal charge distribution changes if it gets into an excited state.