Maybe this question is both in the fields of engineering and physics.
As it seems the electrical quantities like Resistance, Capacitance, Inductance, and so on are quantities we assign to distributed bodies(like solid cylinders, solid cubes, etc). But later on by assuming something named "Lumped element model of electrical components" to be correct, we easily localize these quantities to some single points and then easily draw schematic models of electrical systems with zero diameter lines as wires and pointy entities as resistors or capacitors and then it turns out the calculations are always correct.
My question is:
Is there any mathematical proof to this so called model? I mean how can we assume that for example the resistance of a solid disk is located at a point on (say) its center?
P.S. I think the proof should be in a manner like how we prove the forces exerted on a not rotating rigid body could be considered as just exerted on a point particle at the body's center of mass which has a mass equal to the body's total mass.
Answer
The lumped elements approximation of electrical circuits uses the quasi-stationary approximation for the solution of Maxwell's equations. This means that the speed of electromagnetic field propagation c can be neglected (can be assumed to be infinite). Roughly this means that the dimensions $l$ of the circuit are much smaller than the vacuum wave length $l≪\lambda = c/f$ at the considered frequencies $f$.
A mathematical proof based on retarded potentials solutions of Maxwell's equations can be found, e.g., in chapter 4 of the textbook Ramo, Whinnery, van Duzer, "Fields and Waves in Communication Electronics, John Wiley & Sons Inc., 1994
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