There’s two types of energy associated with a current we should distinguish. Firstly there’s the power flowing through the circuit, then there’s energy associated with having current flowing in a wire at all. So if we’re looking at a piece of extension cord that’s powering a lightbulb, the power flowing through the circuit is what’s making the lightbulb shine. This is governed by the equation P=IV. But there’s also some energy associated with having current flowing in a wire at all. For example, you can work out what the magnetic field should be around a wire with a given amount of current flowing through it and calculate the energy stored in the magnetic field. (This energy is associated with the inductance of the wire.) Similarly, the kinetic energy associated with the electron drift velocity is also there just because the wire has current flowing through it. (This is typically a very small amount of energy.)
To see that these types have to be distinct, think about what happens when we double the voltage going into the extension cord and also double the resistance of the lightbulb it’s powering. Current stays the same, but with twice the voltage we now have twice the power flowing to the light bulb. Because current hasn’t changed, neither has the magnetic field around the wire, nor the drift velocity. So the energy associated with having a current flowing in this wire is unchanged, even though the power provided to the light bulb has doubled. The important thing about the drift velocity in the context of P=IV is that it moves charge. We can calculate the potential energy associated with a charge in a wire as E=qV, and then taking the time derivative gives the power equation. It’s true that drift velocity is also a velocity, and thus the charge carriers have kinetic energy too, but this is not the energy that powers the light bulb.
In terms of exponential attenuation, even DC through resistors gives exponential attenuation if you have a “transmission line” configuration of resistors that look like this:
So exponential attenuation doesn’t seem too unusual or surprising to me.
There’s two types of energy associated with a current we should distinguish. Firstly there’s the power flowing through the circuit, then there’s energy associated with having current flowing in a wire at all. So if we’re looking at a piece of extension cord that’s powering a lightbulb, the power flowing through the circuit is what’s making the lightbulb shine. This is governed by the equation P=IV. But there’s also some energy associated with having current flowing in a wire at all. For example, you can work out what the magnetic field should be around a wire with a given amount of current flowing through it and calculate the energy stored in the magnetic field. (This energy is associated with the inductance of the wire.) Similarly, the kinetic energy associated with the electron drift velocity is also there just because the wire has current flowing through it. (This is typically a very small amount of energy.)
To see that these types have to be distinct, think about what happens when we double the voltage going into the extension cord and also double the resistance of the lightbulb it’s powering. Current stays the same, but with twice the voltage we now have twice the power flowing to the light bulb. Because current hasn’t changed, neither has the magnetic field around the wire, nor the drift velocity. So the energy associated with having a current flowing in this wire is unchanged, even though the power provided to the light bulb has doubled. The important thing about the drift velocity in the context of P=IV is that it moves charge. We can calculate the potential energy associated with a charge in a wire as E=qV, and then taking the time derivative gives the power equation. It’s true that drift velocity is also a velocity, and thus the charge carriers have kinetic energy too, but this is not the energy that powers the light bulb.
In terms of exponential attenuation, even DC through resistors gives exponential attenuation if you have a “transmission line” configuration of resistors that look like this:
So exponential attenuation doesn’t seem too unusual or surprising to me.