Self-consistency check

It is well known that the divergence of Faraday's law (4) is written

$$\frac{\partial \nabla \cdot \mathbf{B}}{\partial t} = - \nabla \cdot (\nabla \times \mathbf{E}) = 0,$$ (9)

which implies that $\nabla \cdot \mathbf{B}= 0$ will hold in later time if it is satisfied at the initial time.

Because the displacement current is neglected in Ampere's law, the divergence of Ampere's law is written

$$\nabla \cdot \mathbf{J}= 0.$$ (10)

On the other hand, the charge density is defined through Poisson's equation, Eq. (7), i.e.,

$$\rho_q \equiv \varepsilon_0 \nabla \cdot \mathbf{E}$$

$$= \varepsilon_0 \nabla \cdot (\eta \mathbf{J}-\mathbf{u} \times \mathbf{B}) .$$ (11)

which indicates that the charge density $\rho_q$ is usually time dependent, i.e., $\partial \rho_q / \partial t \neq 0$. Therefore the charge conservation is not guaranteed in this framework. This inconsistency is obviously due to the fact that we neglect the displacement current $\partial \mathbf{E}/ \partial t$ in Ampere's law. Since, for low frequency phenomena, the displacement current $\partial \mathbf{E}/ \partial t$ term is usually much smaller than the the conducting current $\mathbf{J}$, neglecting the displacement current term induces only small errors in calculating $\mathbf{J}$ by using Eq. (5).