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 $(\mathbf{J}\cdot \mathrm{d}\mathbf{S})\cdot \mathrm{d}t$.