If the electrons leaving and entering the battery per unit time is different during transient phase then how it's voltage doesn't change?

Question

I am a high school student and I am very confused in how battery's and electricity actually works?

My question is: suppose we have an ideal battery, we know as soon as we switch on the DC circuit, a transient current will start to flow and at this moment the current at different parts of the circuit will be different and charges will be depleting and accumulating on different parts, so at this moment the charges leaving the battery and entering the battery would be different and it would only be equal when steady state is reached. So shouldn't the potential difference of the battery get changed because until steady state the charges leaving and entering battery weren't same? In high school Textbooks,it's written that the charges entering and leaving the battery would be same because battery's potential difference is maintained ( the electrons leaving from the Zn rod will end up at the copper rod) but it seems like a flawed statement, battery's doesn't have their own brain. How do they 'know' that at all the time they have to maintain the potential difference?How can the same no. Of charges per unit time will end up at copper rod at all the moment, this should only happen at steady state and this even an ideal battery's potential should change a bit?

Answer 1

The issue you are running into is one of conflicting assumptions. In the high-school textbooks you reference, and indeed in college-level circuits textbooks, they are using the assumptions of circuit theory.

The particular assumption that is causing this issue is the assumption that all electromagnetic effects happen instantaneously, in other words circuit theory assumes that the speed of light is infinite. This assumption is sometimes called the small circuit assumption because it sets a length scale for the circuit, but it more explicitly sets a time scale.

With that assumption we cannot directly investigate the transient behavior you mention at all. We are assuming that we are not interested in such short time scales, so we cannot use that assumption if we are explicitly interested in those time scales.

With that assumption removed it is unclear what assumptions you could logically make about an “ideal battery”. However, for a real battery it is clear that at the transient time scale the voltage may indeed vary and the currents at the two electrodes may indeed differ.

This is not to say that the high-school textbook is wrong. It is just using an assumption that is explicitly violated in your question about the transients.

Answer 2

A few more remarks, to already good ones made.

1. Textbooks do (and have to) make things ideal in the beginning. This is to demonstrate basic concepts. Once you've understood them, the rest will be easier to understand.

2. Charges have no brain. But assume, they would accumulate somewhere ... then this would create an extra electric field, which creates an extra current (=moving charges) UNTIL this accumulation no longer exists. // That's the basic concept of equilibrium. If it's disturbed, it restores itself. Nature.

3. The real world can be approximated step by step. The first action of making the overideal battery a bit more real is introducing an internal ohmic resistance. With this simple model you can already model, understand and reproduce many real-world observations.

4. If you recall Kirchhoffs law a network of resistors simply distributes currents (such that net current remains zero), which is another way (Ohms law) to state a potential distribution accross your network (voltage drops across each resistor). This is the world of "systems of linear equations via U=R I.

5. If you also have capacitors and/or inductors in your network, things become more troublesome. Either you enter the world of systems of differential equations, or, which is equivalent and "easier" to handle, the world of systems of linear equation in the complex variable s from Laplace transformation. // The key here is: when you wait long enough, all time-dependent effects settled down, and you can simplify your network by replacing all C's by open wires, and all L's by short-circuits ... and you are back to #4.

In a nutshell, describing the real world with simplified and idealized components (like L, C, R) needs a good understanding of what is going on. The better you understand, the simpler AND more accurate your model can be.