I know the mathematical reason behind this. Given the inductor I-V characteristic we know that:

I understand mathematically that a function which is differentiable must be continuous, but what is the physical reason behind this?

We know that current creates magnetic field and that the energy stored in that magnetic field is:

I guess it is pretty much related to the energy somehow, but then what's next, and what is the explanation for this?

The same question applies to the voltage of capacitor and energy stored in the electric field in case of a capacitor.

## Best Answer

Here's the real reason.

A moving charge (current) produces a magnetic field.

When a current is applied to an inductor (coils of wire), there is a magnetic field that builds up within the inductor.

The change in magnetic field (flux) produces an EMF across the inductor. The EMF opposes the change in current through the inductor.

So you have an EMF fighting the change in current.

The higher the rate of change in current, the higher the EMF across the inductor opposing it.

That's why the current doesn't change instantaneously.

Key point to remember is that the energy storage mechanism of an inductor is a magnetic field.

The magnetic field (energy) also can't change instantaneously. It's physically impossible to instantaneously change the energy in an inductor (or capacitor).

The fields build up or collapse with respect to time.

Have a look at Lenz's and Faraday's laws. https://en.m.wikipedia.org/wiki/Lenz's_law https://en.m.wikipedia.org/wiki/Faraday's_law_of_induction

One important equation is this:

EMF = -N*dΦ/dt

Look at the EMF equation in terms of magnetic flux.

See that EMF is proportional to the rate of change in flux?

Looks similar to the equation in terms of current doesn't it :-)

Here are some more examples along with a simple picture: https://andropoide.blogspot.com/2017/02/inductor-lenz-law.html?m=1

Hope this helps.