Suppose we have an ideal LC circuit (no resistance) and an open switch where the capacitor has an initial voltage $V_o$. Initially, the energy stored in the capacitor at $t=0$ is $\frac{1}{2}CV_o^2$ and the energy in the magnetic field of the inductor is zero because no current is flowing. Now at time $t=0+dt$ we close the switch and current slowly begins to build up. When the current is a maximum, the energy stored in the magnetic field of the inductor is $\frac{1}{2}LI^2$ but now the energy stored in the capacitor is zero. Thus we must have that $\frac{1}{2}LI^2=\frac{1}{2}CV_o^2$ because no energy is dissipated since there is no resistance.
But there seems to be something very wrong here at a fundamental level. The charge (the electrons) traveling through the inductor at the instant that the current is a maximum have a non-zero kinetic energy (denote this kinetic energy $K_{charge}$). They have to have non-zero kinetic energy since they constitute a current. But if they do posses this energy in addition to the magnetic field energy $\frac{1}{2}LI^2$, then the total energy at the moment the current is a maximum will equal $E_{tot}=\frac{1}{2}LI^2+K_{charge} >E_{initial}=1/2CV_o^2$. So its seems we have created energy in this process?
The only way I can work around this issue is to assume that the kinetic energy is already somehow factored into the magnetic field energy but I am not sure.
Any help on this issue would be most appreciated!