What are the "divergence equations," and how do they follow from the presented Maxwell's equations?

Question

I am told that Maxwell's equations take the form

$$\text{curl} \ \mathbf{E} = - \mu j \omega \mathbf{H}, \ \ \ \ \ \text{curl} \ \mathbf{H} = (\sigma + \epsilon j \omega) \mathbf{E},$$

where $\sigma$ is the conductivity, the dielectric constant is $\kappa_e = \epsilon / \epsilon_0$, the magnetic permeability is $\kappa_m = \mu / \mu_0$, $j = \sqrt{-1}$, and $\omega$ is the angular frequency.

I am then told that "the divergence equations follow automatically from these." What are the "divergence equations," and how do they follow from the above Maxwell's equations?

Answer 1

I guess the only way to find out is to apply a divergence to the equations you mentioned and use the fact that $\nabla\cdot(\nabla\times \vec F)=0$ (i.e., div of curl is zero) for any differentiable vector $\vec F$. You'll end up with $$-j\omega\nabla\cdot(\mu \vec H)=0$$ and $$\nabla\cdot\vec J_{\rm free}+j\omega\nabla\cdot(\epsilon\vec E)=0$$ where I have defined $\vec J_{\rm free}=\sigma\vec E$ (and this holds in a conducting medium). Now define $\vec B=\mu\vec H$ and $\vec D=\epsilon\vec E$ so that the above equations become $$\nabla\cdot \vec B=0$$ and $$\nabla\cdot\vec J_{\rm free}+j\omega\nabla\cdot\vec D=\nabla\cdot\vec J_{\rm free}+j\omega\rho_{\rm free}=0$$ where I have used another divergence equation $\nabla\cdot\vec D=\rho_{\rm free}$. The above equation (for $\vec J_{\rm free}$) is known as the continuity equation.

Note that your equations are in frequency domain, i.e., to convert to time domain you can replace $j\omega$ with $\frac{\partial}{\partial t}$. A note on Maxwell's equations: The bundle of the four equations, i.e., the two curl equations you have in the question along with the two divergence equations for $\vec B$ and $\vec D$ are formally known as the (differential form) of Maxwell's equations. Anything else may be derived from these four. Also note that the curl equation you provided in the question for $\vec H$ holds for a conducting medium. More generally, it is given by $\nabla\times \vec H=\vec J_{\rm free}+\frac{\partial \vec D}{\partial t}$. Finally, $\vec B$ and $\vec D$ are respectively related to $\vec H$ and $\vec E$ via constitutive relations in media as demonstrated in the equations above. Please let me know if you find any errors in this answer or you have further questions.