Using Kirchhoff's rule in RC circuit while charging

[issacnewton]

Hello

I have a conceptual question about using Kirchhoff's voltage rule to derive expression of the charge in the RC series circuit when it is being charged by certain EMF. Usually most books start by writing the equation
$$\mathscr{E}-iR-\frac{q}{C}=0$$

But when the charging is going on, the current is changing and that means there is non-zero rate of change of magnetic field through the loop which makes the circuit. Kirchhoff's voltage rule is derived from the Faraday's law when the rate of change of magnetic field is zero and so the right hand side of the Kirchhoff's rule is zero. But when we are charging the circuit, right hand side would no longer be zero. So how would we conceive the above equation ? When I was reading online notes of Prof. Walter Lewin of MIT, he said that most of the physics books get these things wrong. Here are the relevant links from his website.

http://ocw.mit.edu/courses/physics/...etism-spring-2002/lecture-notes/lecsup315.pdf

http://ocw.mit.edu/courses/physics/...netism-spring-2002/lecture-notes/lecsup41.pdf

So what do people think here ?

thanks

 

[thegreenlaser]

You're right, but the assumption is that this effect is negligible (sometimes it's called the quasi-static approximation). For circuit analysis in the "normal" low-frequency range (e.g. 60 Hz from an outlet in your house) these kind of assumptions are justified. In fact, we make a lot of these kinds of assumptions, and you'll find that they actually start to break down if you look at frequencies in the MHz or GHz range. Then you have to start being a lot more careful about stuff like this (that area of study is called RF or Microwave circuits). For example, if you analysed a circuit like this exactly using Maxwell's equations, you'd probably find that there's actually small amounts of charge building up and dissipating at different points along the wires, which could be a lot of work to model properly. However, at the low frequencies where we usually do circuit analysis, these effects are small enough to be neglected.

If you're really interested in this sort of thing, though, I would suggest taking a course on RF/Microwave circuits if you can (I'm working through one right now)

 

[issacnewton]

Hey greenlaser,

Thanks for the reply. Yes it makes sense. But even at low frequencies like 60 Hz, there could be violations of the assumptions of the Kirchhoff's rules. Look here

Significant violations of KCL can occur even at 60Hz, which is not a very high frequency

But as you said, in most cases probably its very small effect. But for pedagogical reasons, physics books should mention this. It can cause huge amount of confusion. Thats why Prof. Lewin scolds a well known author like Giancoli.

And I would be interested in taking a course in RF/Microwave circuits. Any links ?

thanks
IssacNewton