Question

For a metal sphere, what must the value of the relative dielectric constant be for the polarizability to become infinite? What is this resonance called?

Answer 1

Polarizability α is defined by the induced dipole moment p and the electric field E that causes this moment:

$$\alpha=\frac{p}{E}.$$

We know that the field depends on the dielectric constant:

$$E=\frac{1}{4\pi\epsilon\epsilon_0}\frac{q}{r^2}.$$

Polarizability becomes infinity when field becomes zero. This happens when the relative dielectric constant epsilon becomes infinity (and it is infinity for metals).

However, to answer "What is this resonance called?", some corresponding context is needed.