Quote by Andrew Mason
It can exceed the applied voltage. That is exactly what happens when a DC current through the coil is interrupted. What you cannot have is more power from the induced voltage than from the power source.
When the DC circuit is closed, the applied voltage causes current to flow and the induced backemf causes current to flow in the opposite direction. The rate of change of current cannot produce an induced voltage that causes more current to flow through the resistance than the applied voltage causes to flow. If you could do that you would be producing more energy than you are delivering to the circuit.

That's a good point. I never thought about that one. However, I think I do have a good understanding of the dynamics of the inductor now, having thought about them for a long long time
. I have a picture about how the fields are hindering current to flow. What I'm trying to understand right now is how to understand if the differential equation is complete or if it describes an ideal situation.
My suggestion was that it wasn't complete and that you would need a damping term [itex]k \cdot \frac{di}{dt}[/itex] to make it more realistic. For suppose the electrons were extremely heavy. Then they wouldn't accelerate very easily. And the acceleration of electrons is the same thing as increasing current. Then current would need more time to build up.
Now suppose the electrons are as ultralight as they are in reality. Then they would still hinder current to build up. And the damping term would still be valid, although the constant k would be small. (Having appreciated it in my head it seems to be small.) This applies to the equation [itex]u = R \cdot i[/itex] as well. Otherwise current would build up instantaneously, revealing an infinite force.
You could also need terms to take heat loss into consideration.
To solve this new differential equation could be messy, or it could be just as easy. But it shouldn't be done really. The purpose of the equation is to show that there is a connection between mechanics and electricity. In the original differential equation there is no such connection. Isn't this a valid point? To understand any differential equation you would have to know what it doesn't take into account. This is often left unanswered by physicists. That is my point.
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