Lets now consider the LC circuit in figure 2.3 which has no resistive element.
Kirchoff's voltage law applied to the loop is
Substituting our previous expressions for 
and 
gives
Using I = dQ/dt gives
The circuit equation is second-order in Q and one possible solution is
where 
is the initial charge on the capacitor and 
is an arbitrary phase constant.
Considering the cases of 
, gives 
.
The angular frequency 
is totally determined by the other
parameters of the circuit
and 
is the natural or resonance frequency of the circuit.
We can also solve for the current and voltage across the capacitor
Notice that unlike the transient current and voltage responses of the RC and RL circuits, the LC circuit oscillates. The energy in the circuit is shared back and forth between the inductor and capacitor.
