Range of validity of Kirchhoff's rules

[fluidistic]

I'd like to know if Kirchhoff's laws of voltage and current are valid in circuit with alternate current. If I'm not wrong, in such circuits, the E field provided by the emf is varying with time. And I've heard that Kirchhoff's laws are a special case of Faraday's law of induction but with a constant B(or E?)-field or something like that, I don't remember well.

 

[Pythagorean]

I'd like to know if Kirchhoff's laws of voltage and current are valid in circuit with alternate current. If I'm not wrong, in such circuits, the E field provided by the emf is varying with time. And I've heard that Kirchhoff's laws are a special case of Faraday's law of induction but with a constant B(or E?)-field or something like that, I don't remember well.

K's laws are approximations derived from Maxwell's equations (well, maybe they weren't originally, but they can be derived from them).

 

[fluidistic]

K's laws are approximations derived from Maxwell's equations (well, maybe they weren't originally, but they can be derived from them).

Yes, that's why I said I've heard that it's a special case of Faraday's law of induction. I just wonder when I can apply them or not. For example in AC circuits.
I know that if there's a circuit with a changing magnetic field instead of a common emf, they are not valid anymore.

 

[Meir Achuz]

You can use K's rules with AC if you include L(dI/dt) and Q/C.

 

[fluidistic]

You can use K's rules with AC if you include L(dI/dt) and Q/C.

Wouldn't that be for RLC circuit?
But for a circuit containing an AC emf and a resistor, can I apply Kirchhoff's rule the same way as if the circuit was with DC?

Edit: My doubt arises because of this video: , between minute 39 and 43. (you can pass directly to this moment if you like).

 

[saunderson]

You can easily derive Kirchhoff's rules from the maxwell equations, if you assume that

$$\frac{\partial \vec B}{\partial t} = 0$$
​

then the relevant maxwell equations are

$$\mathrm{rot} \, \vec E(\vec r) = 0 \qquad \mathrm{;} \qquad \mathrm{div} \, \vec B(\vec r) = 0$$
​

Kirchhoff's voltage law is the integral expression of

$$\mathrm{rot} \, E(\vec r)=0 \qquad \Rightarrow \qquad \oint \limits_{\mathcal{C}} \mathrm{d} \vec r \, \vec E(\vec r) = \sum_{i=1}^{N} V_i = 0$$​

So if the magnetic field is time dependent, the equation above isn't valid any longer, because then the relevant maxwell equation is

$$\mathrm{rot} \, E(\vec r,t)= - \frac{\partial \vec B(\vec r,t)}{\partial t} \qquad \Rightarrow \qquad \oint \limits_{\mathcal{C}} \mathrm{d} \vec r \, \vec E(\vec r,t) = \sum_{i=1}^{N} V_i = - \oint \limits_{S} \mathrm{d}\vec S ~ \frac{\partial \vec B(\vec r,t)}{\partial t}$$​

But for small frequencies Kirchhoff's voltage law is still a very good approximation.

 

[fluidistic]

Ok thanks a lot saunderson. I'm still confused about AC circuits. There's no B field (or is there?), but a varying E field.
Also what range of frequencies can we take as small? I guess it depends on the uncertainty I want... ok.

 

[saunderson]

yes the e-field is varying with time! But if the e-field is varying, the b-field must varying with time too.

In the case of time varying fields, maxwell's equations doesn't decouple. Like you see in the expression below, the b-field depends on the time varying e-field and the other way around, too.

$$\mathrm{rot} \, \vec B(\vec r, t) = \mu_0 \vec j(\vec r,t) + \varepsilon_0 \mu_0 \frac{\partial \vec E(\vec r,t)}{\partial t}$$​

So, in the case of time varying fields you mustn't consider only one field. You always have to consider both!

 

[fluidistic]

Thank you very much. You've cleared my doubts.

 

[diazona]

For what it's worth: as far as I know, Kirchoff's laws are a good approximation if the time it takes for an EM wave to propagate through the circuit is much less than the time scale over which the voltage varies (e.g. the period of oscillation of the AC source) Although I've never worked through the math to prove that, so take it with a grain of salt.

 

[bjacoby]

For what it's worth: as far as I know, Kirchoff's laws are a good approximation if the time it takes for an EM wave to propagate through the circuit is much less than the time scale over which the voltage varies (e.g. the period of oscillation of the AC source) Although I've never worked through the math to prove that, so take it with a grain of salt.

Look. If you have some configuration of material (conductors, insulators, magnetic materials etc.) in space and you want to know what happens (currents, forces, etc) when you apply various voltages, fields, etc. a reasonably accurate approximation can theoretically be obtained in most cases using Maxwell's equations. But the problem is that the mathematics of the solutions are so complex that only very simple cases can actually be solved. So to make answers to electric and magnetic phenomena more practical circuit theory was invented. This involves a series of simplifications and assumptions applied to our configurations so as to make solutions feasible. One such assumption is that our configuration (circuit) does not radiate energy. This allows us to use conservation of energy from one part of the circuit to another as none is being lost in the summation. The above rule is just one way of assuming that there is negligible radiation. Another is to make sure the size of your "circuit" is much smaller than the wavelength of the highest frequencies you are applying to it.

Other circuit assumptions are for example that "terminals" exist. In other words if you apply a voltage to two points of the circuit which we call "terminals" we assume that the potential measured at any point on those terminals does not vary with position. Often components are "idealized" where capacitors are assumed to have no inductance and inductors are assumed to have no capacitance. In a nutshell, we take chunks of our material configuration and create simplified "lumped" parameters that more or less represent the essence of a given arrangement of matter like say a coil of wire or some flat plates. The bottom line in all this is that while some accuracy is lost even if assumptions are carefully made, solutions to many practical problems become easy and straightforward which would be impossible even using large computers by field theory.

 

[Andy Resnick]

I'd like to know if Kirchhoff's laws of voltage and current are valid in circuit with alternate current. If I'm not wrong, in such circuits, the E field provided by the emf is varying with time. And I've heard that Kirchhoff's laws are a special case of Faraday's law of induction but with a constant B(or E?)-field or something like that, I don't remember well.

I thought Kirchhoff's law of voltage can be derived from conservation of energy, and the law of current can be derived from the law of conservation of charge.

 

[sophiecentaur]

I think that Kirchhoff's laws might fall down once there is significant energy radiated from the circuit. Another 'resistor' starts to turn up in the form of 'radiation resistance' and you would really need to include it in every element / connecting wire in the circuit. It's ok as long as the circuit is small enough compared with the wavelength because the radiation resistance is vanishingly low.

But, perhaps, K2 should apply as you go round a loop (as long as you include a total description of the impedances of all elements (for the IZs) or else you'd end up with a discontinuity when you got back to where you started from. And, as the extent of a node is zero then K1 should also be followed.

 

[fluidistic]

I thought Kirchhoff's law of voltage can be derived from conservation of energy, and the law of current can be derived from the law of conservation of charge.

I do not know the assumptions it uses. In post #5 I posted a link to a video. From minute 39 and on, we see an example in a circuit where there's no battery, rather a variable magnetic field induces an emf in the circuit. The difference of potential between 2 points depends on the path taken in the circuit. So going through a resistor on say the right side, is different from going to the left side and going through 2 resistors. Hence it seems there are 2 values as of difference of potential between 2 points. Kirchhoff's laws cannot explain this. So there's something Kirchhoff assumes that cannot be assumed in the case of a changing magnetic field instead of an emf in a circuit.

 

[sophiecentaur]

Is it really invalid to describe the induced emf as just an emf which can be used in K2?
I can't see why. It doesn't necessarily have to be from a battery.

 

[sophiecentaur]

To calculate the result for a general circuit, you would need to know the separate values of emf, generated in each loop (or, possibly each section of each loop) of the circuit and put these values into the K2 calculation. I really don't think there should be any objection to that - as long as you were able to calculate the induced emfs (the difficult bit).