Do kinetic energy and inductor energy violate conservation of energy?

Question

I posted this question on the physics page. I feel like it might be better suited to this site, though.

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.

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 it 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.

Answer 1

Voltage and current are emergent approximations, not fundamental things. If you want to bring the velocities of electrons into consideration, then you have to understand what's going on at the quantum mechanical level. Which is hard work.

Working with I and V requires ignoring the electron behaviour, just as working with pressure, which is also an emergent approximation, requires ignoring individual atoms' velocities and sizes. While pressure and volume are well approximated over a wide range of values, the finite size of atoms causes pressure to deviate from the simple linear laws at very high pressure, and the finite velocities cause (for instance) muzzle velocities to be limited in guns that use expanding gases as a driver.

It turns out that using I and V as if they are fundamental works very well over many more orders of magnitude than do the gas laws approximate the behaviour of gases.

In practice, the L is inferred from energy measurements, so you could consider that any small electronic contribution to energy is already accounted for.

Answer 2

In addition to the other answers, I would like to add that there are two different kinds of inductance.

1. is the usual inductance that describes the energy stored in the magnetic field.

2. is the kinetic inductance. This happens whenever the mean free path of electrons becomes appreciable, e.g. when transversing vacuum, in superconductors or some other materials which have topological reinforcement of conducting states leading to ballistic transport. This contribution to inductance accounts for the inertia of the charge carriers. An impressive demonstration of this is trying to change the current through a superconductor.

Both are inductances because they describe the desire for a current to continue flowing. When using a component with inductance L in electrical engineering, it does not disclose the origin of its inductance.