Is Energy Conservation Preserved in a Circuit with Two Inductors?

[Rolls With Slipping]

Homework Statement

This is my own question I created to try and think about what would happen if two inductors with different currents are connected.

Assume the circuit elements all have non-zero values. The switch has been closed for a very long time such that the current through $L_1$ is constant and no current flows through $L_2$. What will the current be through $L_1$ and $L_2$ after opening the switch?

Relevant Equations

$V_L = L \frac{dI}{dt}$

The standard assumption is that the current through an inductor must be continuous such that you don't produce an infinite back emf. However in this case, the current through $L_1$ is finite before opening the switch and the current through $L_2$ is zero before opening the switch. When the switch opens, is it possible for the current to be continuous through both?

 

[DaveE]

If you assume ideal components, then there is no finite solution. An instantaneous change in inductor current would require infinite voltage. In practice you would probably create an arc across the switch contacts.

 

[DaveE]

However, this circuit also shows what is fundamentally wrong with the idealized lumped element circuit models. You can not actually build a perfect inductor, it will have some capacitance. This is how classical EM things are. There is always inductance from moving electrons and capacitance from generated voltages.

 

[tech99]

What you depict seems to be just a resonant LC circuit. It has L, C and R. After a long time with the switch closed, C charges to the battery voltage. When the switch opens, a damped oscillation occurs which gradually decays, depending on the circuit constants.
As you will know, an inductor stores energy and behaves like a heavy flywheel or mass - it is hard to start it and hard to stop it. In a more general case where two inductors carrying current are suddenly connected together, I think it is analogous to a collision between masses. For an inductor, the stored energy is LI^2/2, and for a mass the KE is mv^2/2. The momentum of the mass is mv and the "momentum" of the inductor is LI. So by analogy, I suggest that for two inductors, after the "collision", momentum LI is conserved, such that L1I1 + L2I2 = (L1 + L2) I3. This involves dissipation of energy.