What is the difference between the dipole in IR and Raman vibrations? In order for a vibration to be IR active, there must be a change in the dipole moment ($\mu$) of the molecule, which is given by:
$\mu$ = $\sum$ q * r
Where q is the charge magnitude and r is the distance between the two charges. 
Now, whenever I see a classical description of Raman spectroscopy, it starts off by describing the formation of an induced dipole in a molecule due to an electromagnetic field:
$\mu$ = $\alpha$ * E
Where $\alpha$ is the polarizability and E is the magnitude of the electromagnetic field. 
When the E field oscillates, it also causes the induced dipole to oscillate. Since we have an oscillating dipole, how is this not IR active? 
What is the difference between the induced dipole occurring in the IR example vs the Raman example? If someone could describe this answer in terms of oscillating electron and protons, I think it might help as well.
 A: Both IR absorption/Rayleigh scattering and Raman scattering involve oscillating electric moment of molecules. But just because there is Raman scattering, and therefore electric moment oscillates, it does not mean that the excited mode has large transition dipole moment (which I think is what you mean by "IR active").
Transition dipole moment is a matrix element characterizing pair of Hamiltonian eigenfunctions. It can be zero for some pair and nonzero for other. If it is large, large absorption / scattering (if molecules are very rarified) at that pair frequency is to be expected. 
But Raman scattering can happen even if that transition dipole moment is low or zero. This is because the electric dipole oscillations are then proportional to polarizability matrix element, not just the electric dipole matrix element for that transition. This polarizability can be high even if the dipole moment matrix element is low(zero). Mathematically, polarizability depends on many transition dipole moment elements for all possible pairs of states, not just the one that is implied by the radiation frequency. See for example the formulae 2.55, 2.62 in  https://application.wiley-vch.de/books/sample/3527405062_c01.pdf on the page 21.
