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When dealing with AC or RF circuits, does one need to take into account the displacement currents (as derived from Maxwell's Laws or Heaviside's formulation) when applying Kirchhoff's law to a circuit analysis?

Why (or when) is this not normally done in circuit analysis tutorials?

hotpaw2
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  • You need to, and you do "take into account the displacement currents": otherwise, capacitors and microstrip stubs would participate in calculations as circuit breakers and not as useful reactive components. – V.V.T May 21 '21 at 01:57
  • Interesting. It appears that most tutorials on A/C circuit analysis do not mention the term "displacement current" anywhere. Maybe it's called something else? – hotpaw2 May 21 '21 at 03:25
  • So most circuit analysis tutorials do not mention displacement currents with respect to Kirchhoff's law because of their failure to clearly state that the textbook chapter only covers circuits consisting of lumped element approximations to reality? – hotpaw2 May 21 '21 at 14:31

2 Answers2

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Displacement currents inside most circuits are limited to the space between capacitor plates (AFAIK). Outside circuits, the EM waves have changing electric field, so they too involve displacement current.

Kirchhoff's laws are based on the approximations / simplification made based on conservation of Energy (KVL) and conservation of charge (KCL) when applied to lumped element approximations of circuits.

[Wikipedia]

Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits.

In lumped parameter models, the capacitor is a lumped device, i.e., a point and so the gap between the plates are not modelled. They also do not deal with EM radiations (AFAIK). So both opportunities to use the concept of displacement current do not exist in those circuits which are modelled by lumped element models.

AJN
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Does Kirchhoff's law apply to displacement currents?

Kirchhoff's Current Law is exact if and only if you include displacement currents, (or displacement currents are not present).

Kirchhoff's Current Law follows immediately from Heaviside's formulation of Ampere's Circuital Law with Maxwell's addition of a displacement current term.

$$\nabla \times \vec{B} = \mu_0(\vec{J} + \epsilon_0\frac{\partial\vec{E}}{\partial t})$$

where \$\vec{J}\$ is the conduction current density, and \$\epsilon_0\frac{\partial\vec{E}}{\partial t}\$ is the displacement current density.

Taking the divergence of both sides (and cancelling out \$\mu_0\$ gives Kirchhoff's Current Law in microscopic form.

$$0 = \nabla \cdot (\vec{J} + \epsilon_0\frac{\partial\vec{E}}{\partial t})$$

In macroscopic form, this says that in any closed region in space, not just in a circuit, and not just with "lumped elements", the sum of all conduction currents and displacement currents into (or out of) that region is exactly 0.

If Maxwell's equations hold, then Kirchhoff's Current Law (with displacement current) holds, without exception, and exactly.

Detailed Discussion of Kirchhoff's law and displacement current point and its implications are found in:

  1. Eisenberg, R. S. 2019. Kirchhoff's Law can be Exact. arXiv preprint available at https://arxiv.org/abs/1905.13574.
  2. Eisenberg, B., N. Gold, Z. Song, and H. Huang. 2018. What Current Flows Through a Resistor? arXiv preprint arXiv:1805.04814.
Voltage Spike
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Math Keeps Me Busy
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  • Is there a similar extension to KVL ? – AJN May 21 '21 at 15:49
  • @AJN Regarding KVL, it is exact when expressed in the proper form and inductive emf is properly defined. I will find a link for you. – Math Keeps Me Busy May 21 '21 at 15:55
  • I think I understand what you mean. any energy the charge gains when traveling once around a loop is then balanced by placing the necessary induced EMF in the loop ? – AJN May 21 '21 at 15:59
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    @AJN KCL is a _node_ law, not a loop law. It says that if you enclose some volume, say a solder blob, by a surface, the total current into that volume (where current "into" can be negative, i.e. out of) is 0. Or, using schematics, the total current into a node (where current "into" can be negative) is 0. KVL is a _loop_ law, and requires one to pick a loop to analyse. [Here](https://electronics.stackexchange.com/a/561968/268467) is an answer I gave regarding the use of KVL. – Math Keeps Me Busy May 21 '21 at 16:46
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    @AJN [Here](https://physics.stackexchange.com/a/631405/290970) is another answer which may help to clarify KVL. – Math Keeps Me Busy May 21 '21 at 16:54