What scale are Maxwell's Equations valid? Both Maxwell's equations and quantum mechanics are used to describe the behavior of electrons in circuits.
I am confused on the interlinking between the two and the dividing line between when you use one vs the other. When must you abandon Maxwell's equations and move to quantum mechanical model when analyzing electron behavior in homogeneous media? At what scale do Maxwell's equations "fail"?
 A: Two of Maxwell's equations assume a "continuity" in electricity where there is in fact "quantization".
Gauss's Law
$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$
and Ampere's Circuital Law with Maxwell's correction
$$\nabla \times \vec{B} = \mu_0 \left(\vec{J} + \varepsilon_0\frac{\partial\vec{E}}{\partial t}\right)$$
refer to $\rho$ (charge density) and $\vec{J}$ current density.
On a atomic/microscopic scale, however, these terms are problematic, as electric charge and electric current are quantized. Therefore, on a microscopic level, quantum electro-dynamics must be used.
A: Quick clarification, based on some of the comments. This answer focuses on where Maxwell's equations are invalid (the title asks when Maxwell's equations are valid, but the end of the post asks when Maxwell's equations are invalid). Second, I usually consider Maxwell's equations to tell us about electromagnetic interactions, and so quantizing Maxwell's equations involves situations where photons are important. There are many situations where quantum mechanics is important in describing the motion of charged matter like electrons and ions, but where the electromagnetic field itself can be considered a classical field. For example, when describing conductivity in various kinds of materials, which is a major topic in condensed matter physics, the applied field generating a current can usually be considered a classical field, but quantum properties of electrons in a periodic potential are very important in determining the band structure, and therefore whether a material conducts or not.
Anyway, two important examples where Maxwell's equations are invalid would be:
(a) When the number of photons is small (for example, you can't use Maxwell's equations to explain the photoelectric effect for low intensities / small numbers of photons).
(b) When describing the interactions of a charged particle whose Compton wavelength is comparable to or larger than the distance scale (or, equivalently, energy scale) of the process you are considering. For example, you don't need QED to describe the interactions of a piece of wool with static charge (the wool has a lot of mass and a tiny Compton wavelength). You don't need QED to describe the repulsion of two electrons that are far apart from each other, compared to the localization of their wavefunction. You do need QED to describe high-energy scattering of electrons where the electrons have enough energy that their wavefunctions overlap, such as in Compton scattering.
