I am interested in what happens between the plates of the capacitor ... I am just trying to understand the theory behind it.
I am assuming that when you say "tank circuit" that you are talking about L in parallel with C; that's the circuit this term is normally applied to.
Assume C is charged to some voltage, V, that L and C are perfect components, and there is no resistance in the circuit. Then imagine a see-saw.
C has parallel plates which are electrically insulated from one another, but there is one plate full of negative charge (tons of electrons) and the other plate is full of positive charge (tons of atoms with missing electrons). If C were not connected to anything, there would be no path for current to flow, and an electrostatic attraction between the opposite charges on either side of the center insulator. So the electrons can't flow through the insulator to the positive plate, but they feel the opposite charge pulling on them.
You now connect an L across the cap; the coiled wire of the L is still a path for the electrons to flow off of the negative plate to the positive charge they are drawn to. So, C discharges through L, and all the electrons rush around to the other plate.
You know how reactive components can't instantly change their charge? That is, there is a ramp-up/down of voltage and/or current? That effect is present here. The inductance of L keeps the electrons from moving all at one instant (current starts at zero and rises according to the time constant of the circuit). Further, the movement of the electrons constitutes current; inductors oppose change of current, even after all the electrons from the negative plate fill all the holes in the positive plate.
So there is an instant when the current through L is at a maximum, and the net charge on each plate is zero; but the inductance keeps the current flowing until electrons are displaced from the previously negative plate to the previously positive plate. Eventually, the energy of the flowing electrons, which was stored in L's magnetic field is released, and has been used to move electrons to the from one plate of the C to the other.
Now, all the energy which was previously stored in L's magnetic field is now stored in C's electrostatic field, and the position of which plate is + and - has switched. At this point, current is zero and voltage is at a maximum.
If both components are perfect, and there is zero resistance in the circuit, there is no energy wasted from the circuit, and no energy needs to be supplied to the circuit. The charge keeps flip-flopping from one plate to the other, at a speed determined by the values of C and L. All that is needed is an initial charge on C.
Real circuits have resistance; therefore, the oscillation starts at a certain level and dies away rapidly. How rapidly depends on how much R there is, which defines the Q of the circuit. So real circuits find some way to inject energy into the circuit, if a sustained oscillation is desired.
Picture that see-saw again. If the center fulcrum is so rusted it's seized up, that's like very high R. If the fulcrum were perfectly frictionless, and each person on the see-saw held a bungee cord suspended from a ceiling, so that upward motion on one side stored energy in the bungee on the other side, that would approximate a tank with perfect components. The a.c. injected in the real circuit should be the same frequency as the desired oscillation; imagine the two ends of the see-saw trying to move up at the same time, or at least out-of-sync... it just don't work.
You still haven't said what you're trying to apply this to. Tank circuits probably make good oscillators at RF; real components needed don't make sense at audio frequencies, compared to other kinds of oscillator circuits.