# Instantaneous current and average current

Imagine a simple electrical circuit with a battery and resistor.

We know that the average current in any part of this circuit remains same .

My question is : is the instantaneous current also same throughout the circuit? And why?

is the instantaneous current also same throughout the circuit?

Yes. For most practical purposes, the instantaneous current is the same throughout a simple (unbranched) circuit.

And why?

The current density at any point in a conductor is given by the microscopic version of Ohm's Law:

$$\vec{J}=\sigma\vec{E}$$

where $$\vec{J}$$ is the current density, $$\sigma$$ is the conductivity, and $$\vec{E}$$ is the electric field.

If the electric field is arranged in such a way that the current along the circuit is different from one cross section to another, then electrons (or holes) will accumulate in those parts of the circuit where there are changes in current. (Like water behind a dam.). The accumulation of charge in those parts of the circuit will change the electric field, and change it in such a way that the current becomes uniform.

If you are dealing with signals that change slowly enough, you can ignore the time when the current is not uniform. If you are dealing with very long lengths, or very fast changes, you need to be aware that the electric field takes a finite time to propagate.

Not really, because of shot noise (due to electric charge localized on particles and discretized into multiples of $$e$$) and thermal noise (due to random thermal motions of those particles).

In your example, on the resistor, average current is

$$\langle I \rangle = \frac{\langle V \rangle}{R}$$ where $$\langle V \rangle$$ is average voltage on the resistor, non-zero due to source of emf in the circuit. And we assume this formula as entirely correct with the usual model used to analyse of DC/AC circuits where we can neglect those random fluctuations.

But in a real circuit, instantaneous value of both current and voltage fluctuate randomly around these average values.