Reduction of Maxwell's equations to classical circuit theory

Can classical circuit theory based on lumped element models be obtained from Maxwell's equations as a limiting case in an appropriate sense?

If this is the case, what exactly are all the assumptions required to reduce Maxwell's equations to circuit theory? What are some good references with a detailed discussion of this relationship?

Classical circuit theory boils down to Kirchhoff’s laws:

KVL: Kirchhoff's voltage law

A.k.a., conservation of energy.

The algebraic sum of all the potential differences around the loop must be equal to zero: $$\sum_i V_i = 0.$$

This comes from Maxwell's third equation:

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \quad \Longleftrightarrow \quad \oint \mathbf{E}\cdot\mathrm{d}\boldsymbol{\ell} = -\frac{\mathrm{d} \Phi_{\mathbf{B}}}{\mathrm{d}t},$$ where $$\Phi_{\mathbf{B}} = \int\int \mathbf{B}\cdot \mathrm{d}\mathbf{S}$$ is the magnetic flux.

Because $$\mathbf{E}=-\nabla V$$, $$V$$ being the potential: $$\oint \mathbf{E}\cdot\mathrm{d}\boldsymbol{\ell} = \sum_{\mathrm{whole\,\,circuit}} V.$$

In electrostatics, $$\mathrm{d}/\mathrm{d}t = 0$$, so $$\oint \mathbf{E}\cdot\mathrm{d}\boldsymbol{\ell} = \sum_{\mathrm{whole\,\,circuit}} V =0,$$ which is KVL.

Induction

In electrodynamics, $$\mathrm{d}/\mathrm{d}t \neq 0$$ so: $$\oint \mathbf{E}\cdot\mathrm{d}\boldsymbol{\ell} = \sum_{\mathrm{whole\,\,circuit}} V = -\frac{\mathrm{d} \Phi_{\mathbf{B}}}{\mathrm{d}t},$$ where you would call the net effective voltage $$V = \epsilon$$ the electromotive force:

$$\epsilon = -\frac{\mathrm{d} \Phi_{\mathbf{B}}}{\mathrm{d}t}.$$ The minus sign is referred to as Lenz's law.

KCL: Kirchhoff's current law

A.k.a., conservation of charge.

The algebraic sum of ALL the currents entering and leaving a junction must be equal to zero as: $$\sum I_{\mathrm{in}} = \sum I_{\mathrm{out}}.$$

This is essentially equivalent to the continuity equation in electromagnetism.

From Maxwell's fourth equation:

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \frac{\partial \mathbf{E}}{\partial t},$$ and taking its divergence:

$$\nabla \cdot (\nabla \times \mathbf{B}) = \mu_0 \nabla \cdot\mathbf{J} + \mu_0\epsilon_0\frac{\partial (\nabla \cdot \mathbf{E})}{\partial t},$$ where $$\nabla \cdot (\nabla \times \mathbf{B}) = 0$$ and $$\nabla \cdot \mathbf{E} = \rho/\epsilon_0$$ because of Maxwell's first equation.

So:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0,$$ which is the continuity equation between the charge density $$\rho$$ and current $$\mathbf{J}$$.

In the integral form, this becomes:

$$\frac{\mathrm{d}Q}{\mathrm{d}t} = - \int \int \mathbf{J} \cdot \mathrm{d}\mathbf{S},$$ where $$Q$$ is the total charge in the volume bound by the surface $$\mathbf{S}$$. This means that a change in charge $$\mathrm{d}Q$$ requires an inflow of charge $$(\mathbf{J}\cdot \mathrm{d}\mathbf{S})\cdot \mathrm{d}t$$.

• The KVL / KCL derivations look nice for resistive circuits, but what about e.g. RLC circuits where $d/dt \neq 0$? – user Mar 24 at 1:31
• Even in those circuits you always start off with KVL, in the same way as you use Newton II to solve for the dynamics of a mechanical system. – SuperCiocia Mar 24 at 1:38