Assumptions involved in circuit derivations

Question

When deriving formulae for RC, RLC, LC circuits, etc., we typically assume that current is steady. Griffiths gave a good explanation of why this assumption is reasonable (he states essentially that nonsteady current forces itself to become steady through electric forces). However, it seems that we also assume a non changing magnetic field when we apply “Kirchhoff's loop rule.” But current of course varies, so the magnetic field changes, and thus the electric field has some curl. So I’m not sure why this assumption is justified. Is the curl of the field negligible, and does the rule hold true practically? If so, why? Edit: I’ve realized that we’re essentially assuming the circuit itself has no inductance; all inductance comes from an inductor that could be present. In addition, the induced forces act only within the inductor itself (so that the voltage drop across the inductor is the emf). Am I correct in this?

Answer 1

Kirchhoff's circuit laws can be derived from general electrodynamics together with the approximation called Lumped-element model for electrical systems.

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For deriving Kirchhoff's voltage law we start with Faraday's law of induction $$\oint_{\partial S}\mathbf{E}\cdot d\mathbf{l} =-\frac{d}{dt}\int_S\mathbf{B}\cdot d\mathbf{S}$$ or $$-\sum_i V_i = -\frac{d\Phi_B}{dt}$$ This is not quite Kirchhoff's voltage law because the right-hand side usually is not zero. But we can impose the first constraint from the Lumped-element model for electrical systems

1. The change of the magnetic flux in time outside a conductor is neglectable. $$\frac{d\Phi_B}{dt}\approx 0$$

(Most importantly this constraint restricts the design of coils so that the magnetic field is confined within the coils and very small outside.)

When this approximation is valid we finally arrive at Kirchhoff's voltage rule. $$\sum_i V_i \approx 0$$

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For deriving Kirchhoff's current law we start with charge conservation $$\oint_{\partial V}\mathbf{J}\cdot d\mathbf{S} =-\frac{d}{dt}\int_V \rho\ dV$$ or $$\sum_k I_k = -\frac{dQ}{dt}$$ This is not quite Kirchhoff's current rule because the right-hand side usually is not zero. But we can impose the second constraint from the Lumped-element model for electrical systems.

2. The change of the charge in time inside conducting elements is neglectable. $$\frac{dQ}{dt}\approx 0$$

(Most importantly this constraint restricts the design of capacitors so that the positive and negative charges within a capacitor cancel each other.)

When this approximation is valid we finally arrive at Kirchhoff's current rule. $$\sum_k I_k \approx 0$$

Answer 2

All of the Kirchoff's laws and nodal analysis are approximations to Maxwell's Equations that sometimes work better than other. They are good approximations when the physical size of the circuit divided by the wavelength of the signals you are looking is very small. When circuit elements start approaching the size of a wavelength you need to use Maxwell's Equations.

I'm not sure you mean that the current is assumed to be steady (maybe steady state)? But L and C are really only interesting for AC signals. C becomes an open and L a short for DC.

Answer 3

However, it seems that we also assume a non changing magnetic field when we apply “Kirchhoff's loop rule.” But current of course varies, so the magnetic field changes, and thus the electric field has some curl. So I’m not sure why this assumption is justified. Is the curl of the field negligible, and does the rule hold true practically?

The original Kirchoff's second circuital law applies even in such cases, and even if the induced field is due to external sources. It is the most general statement of the law, and can account for all kinds of electromotive forces, including batteries. It states

$$ \text{Sum of all electromotive forces acting on current in the chosen closed path} = \text{sum of terms}~R_kI_k~\text{for all elements}~k~\text{in the path} $$ where $R_k$ is resistance of the $k-$th element and $I_k$ is current flowing through it.

So if there is induced electric field near the circuit (even with non-zero curl), it may produce additional emf, and this is taken into account by including this emf in the sum on the left-hand side.

The modern "Kirchhoff's voltage law" (KVL) is a different statement involving drops of potentials in a closed path, not electromotive forces (emfs). Students are taught KVL instead of the original Kirchhoff's formulation for unclear reasons, and this leaves students with incomplete understanding. But perhaps some reasons are that using KVL and drops of potential is 1) close to enginnering practice, where drop of potential can be measured using oscilloscope (while emf is hard to measure), and 2) it provides for a practical and simple algorithm to write down the circuital equations that works in the most usual case (no external emfs). KVL states

$$ \text{Sum of electric potential drops on all elements in a closed path} = 0. $$

This is very different from Kirchhoff's formulation, but is also valid in general, because electric potential at any single time is a single-valued function of position, so it is a mathematical necessity that sum of changes when going in closed path is zero. Thus KVL is not a physical law; when solving the circuits, other facts used often are physical laws, such as the fact that potential drop on a resistor is $RI$, or the fact that potential drop on a perfect inductor is $LdI/dt$. These laws are limited to that familiar restriction that there must not be any external induced electric field acting on current in these elements, the only allowed induced field has to be due to a part of the circuit acting on itself (e.g. an inductor whose induced electric field action is limited to coils of the inductor only).

Answer 4

Inductance is a property of the conducting wire just like capacitance is a property of a conductor. For a general wire which may be stretched out very long, the actual area it encloses in the loop is so large that inductance is negligible. When we create these circuits, we essentially make this assumption but in practical scenarios, this does somewhat have to be accounted for.