# Relating Maxwell Equation to circuit theory?

As we know, circuit theory makes many assumption so that Maxwell equations works out nearly. Some of the assumption are regarding flux(both magnetic and electric) being zero. In cases where we can't make the flux assumption, we (kind of) introduce correction by introducing elements like capacitor and inductor which accounts for magnetic and electric flux in the circuit. Is this the correct way of thinking.

Also if that's the case, then which law does resistance correct for in a circuit (since we assume wires are resistance-free, we compensate it with some internal resistance).

Also there are 4 Maxwell's laws, so I anticipate another component apart from LCR which would be correction for the remaining Maxwell's law.

• Does this answer your question? Commented Oct 8, 2021 at 22:49
• @SuperCiocia No, not entirely Commented Oct 8, 2021 at 23:10
• how does introducing capacitors relate to correlations? What kind of correlations are you talking about? Commented Oct 8, 2021 at 23:13

Also if that's the case, then which law does resistance correct for in a circuit (since we assume wires are resistance-free, we compensate it with some internal resistance).

Resistance, and bulk resistivity, are phenomena related to matter absorbing energy in the form of electrical potential and converting it to heat. As such it is a function of temperature. Resistance in this sense is unrelated to Maxwell's laws (however, we learn about the characteristic impedance through Maxwell's equations which includes the resistor circuit model, $$R$$).

Maxwell's equations tell us about the fundamental relationships between electric and magnetic fields, as well as their sources. Your expectation that there is a circuit component for each law is not physically logical. We have KCL and KVL which are easily spotted from the integral form of the equations (see the Phys SE post noted by @SuperCiocia), and then we have load modeling.

Load modeling is taught in undergrad physics and electrical engineering as a linear combination of resistance ($$R$$), inductance ($$L$$) and capacitance ($$C$$). As I am currently learning in my grad program, load modelling in practice is more complicated than this and these circuit components are just ideal building blocks.

As an example, there might be a node on a power system which has a constant complex power, which is a mathematical simplification of the general power function in steady state AC circuits:

$$p(t)=V_{rms}I_{rms}\cos\Delta\theta(1+\cos2\omega t)+V_{rms}I_{rms}\sin\Delta\theta\sin2\omega t=\langle p \rangle(1+\cos2\omega t)+Q\sin2\omega t$$

This can be broken into its average component, $$\langle p \rangle = P$$, and a time-varying component which does not contribute to the average power, called reactive power or $$Q$$.

These loads exist, and can be realized with physical devices (look up reactive power compensation), but are not strictly described by ohm's law. In phasor notation:

$$S=VI^*=(IZ)I^*=Z|I|^2=P+jQ=constant$$

In order for the voltage and current to provide this $$P$$ and $$Q$$ balancing, the actual load impedance must be variable (there's no guarantee that the voltage is fixed at such a bus). There's no constant impedance model that can yield this voltage-current behavior. Despite this, Maxwell's laws are still valid when analyzing these voltage and currents on transmission lines in a system which has such a node.