Maxwell's equations are a set of four equations for electricity and magnetism, which were considered separate phenomena before. But with Maxwell's equations, we found that they are related dynamically, so these four equations are called electrodynamics. Although they were intended to complete the framework of phenomena related to electricity and magnetism and derive new results like electromagnetic waves as a form of radiation, there are still things outside of it. The major and profound drawback of Maxwell's equations is that they don't include the continuity equation, and thus they are shackled by non-conservation of charge and energy, which means things could appear or disappear without account. Even with this fundamental drawback, it was rejoiced as a successful theory, and many preferred it over the established laws of mechanics, which stunted mechanics in its classical form. In Maxwell's framework of electrodynamics, we need four equations of Maxwell and Ohm's law, which is a form of continuity equation, and is extensively used in practical applications and the use of electricity. Some use the Lorentz force as a fifth equation and ignore Ohm's law because Ohm's law shatters the physics developed on the superiority of Maxwell's equations. However, the Lorentz equation can be derived from Maxwell's equations of divergence and curl of the electric field. So, to complete Maxwell's electrodynamics, we need five equations, including Ohm's law.
Here are the improved Maxwell’s equations, which incorporate the continuity equation and provide a complete and elegant framework for electrodynamics in a set of four dynamical equations in differential form.
$\mathbf{\nabla}\cdot\mathbf{D}=-\dfrac{\partial\rho}{\partial t}\qquad\quad(\text{Ohm's law})\tag1$
$\mathbf{\nabla}\cdot\mathbf{H}=\dfrac{\partial\rho}{\partial t}\qquad\quad(\text{Ampere's law})\tag2$
$\mathbf{\nabla}\times\mathbf{D}=\mathbf{H}\qquad\quad(\text{Faraday's law})\tag3$
$\mathbf{\nabla}\times\mathbf{H}=\mathbf{D}\qquad\quad(\text{Maxwell's law})\tag4$
Equation $(1)$ is Ohm’s law, which ensures that our charges and energy are preserved. Equation$(2)$ is Ampere’s law, but it may shock the reader as it has divergence, implying existence of magnetic monopoles and shows that there is no need of opposite charges for zero divergence. Since $\mathbf{J}=\mathbf{D}$ is Ohm’s law, derived from $(1)$, we can rewrite $(2)$ as $\mathbf{H}=-\mathbf{J}$, the magnetic version of Ohm’s law. Some changes made to Faraday’s law $(3)$ so it can remain compatible. Finally $(4)$ is named as Maxwell’s law because it’s introduced by him to correct Ampere’s law. From $\mathbf{J}=\mathbf{D}$ and $\mathbf{H}=-\mathbf{J}$ we have $\mathbf{D}=-\mathbf{H}$. This shows that how electric and magnetic fields are opposite but equal except for material properties. The current density $\mathbf{J}$ is the cause of both fields, therefore it is incorrect to state that electric and magnetic fields exist in regions free of charges or in free space.
From identity of curl of a curl,
$\mathbf{\nabla}\times\mathbf{\nabla}=\mathbf{\nabla}\cdot(\mathbf{\nabla})-\mathbf{\nabla}(\mathbf{\nabla}\cdot)=-\mathbf{\nabla}^2\tag5$
Taking curl of $(3)$ and using $(4)$ and $(5)$ in derived expression and then re-arranging,
$\mathbf{\nabla}^2\mathbf{D}=-\mathbf{D}\tag6$
Similarly taking curl of $(4)$ and using $(3)$ and $(5)$,
$\mathbf{\nabla}^2\mathbf{H}=-\mathbf{H}\tag7$
As we see, $(6)$ and $(7)$ are vector Laplace or vector Helmoltz equations $\mathbf{\nabla}^2\mathbf{F}+k^2\mathbf{F}=0$, where $k=1$ and $\mathbf{F}=\mathbf{D},\mathbf{H}$. Their possible solutions are harmonic in nature, thus either in sinusoidal form or in imaginary exponential form. Whatever they are, either $\sin(\text{k}\cdot\text{r})$ or $\exp i(\text{k}\cdot\text{r})$, the term in paranthesis remain constant unless some operation is performed upon, this constant term is known as phase. Now the function $\mathbf{F}(\text{r})$ is changing with different values of $\text{r}$ as we move from one point to another in space. This means that the phase $\text{k}\cdot\text{r}$ is also changing, but in a way that's consistent with the condition of phase constancy. The change in the phase is through $\text{r}(t)$ that must be accompanied by a rate of change $\text{v}$, velocity. So, to maintain the condition of phase constancy, we need to change $\text{r}$ to $\text{r}\pm\text{v}t$, where $\pm$ are for forward or backward moving function. This is because the phase $\text{k}\cdot\text{r}$ is effectively advancing at a rate $\text{v}$ in the direction of $\text{k}$. The phase look as $\text{k}\cdot(\text{r}-\text{v}t)$, but $\text{k}=1$ in our case, now we can write $\text{k}=1=\omega/c$. This means any change in the function is through speed or frequency and both are covariant. So the solution of our equation can be, $\mathbf{F}=F_0\sin(\text{k}\cdot\text{r}-v/c\,\omega t)$, as $\text{k}=1$ then $\text{k}\cdot\text{r}=r$ and phase is $r-v/c\,\omega t$. The relation between $v$ and $c$ is $v=c\cos(\theta)$, that makes $\omega'=\omega\cos(\theta)$. Any change in speed is accompanied by a corresponding change in frequency, whether due to a change in medium or direction. Since the speed of a wave is changed by diffraction, the intensity of the diffracted wave is less. The limit of the cosine of the angle is such that the $v$ is less than or equal to $c$. The maximum angle subtended is $\pi$ in a plane and $2\pi$ in space. This limit of the cosine is not applicable to relative motion, but due to the limit of the medium, a wave can diverge at high relative speeds. This might be the cause for the introduction of length contraction.
Maxwell introduced a correction to Ampere's law not because it had issue with conservation laws but to preserve both Gauss's laws and Faraday's law from inconsistency. The original Ampere's law was already consistent with the continuity equation, but Maxwell added a term to make it symmetric with Faraday's law, which allowed him to derive the wave equations. By doing so, Maxwell ensured that the divergence of the magnetic field remains zero, satisfying Gauss's law, and also maintained the consistency of Faraday's law. Additionally, Maxwell's correction enabled him to equate the zero charge and current densities so that electromagnetic wave with property of self sustaining and can travel throgh void become possible. Maxwell's correction was not due to mathematical constraints but conditions were developed later. Maxwell took the $\operatorname{div}$of Ampere's Law which had a non-zero value: $\mathbf{\nabla}\cdot(\mathbf{\nabla}\times\mathbf{B})=\mathbf{\nabla}\cdot(\mu\mathbf{J})=-\mu\partial\rho/\partial t$.To correct this, he introduced a term to make the result zero: replacing $\mathbf{J}$ with $\partial\mathbf{D}/\partial t+\mathbf{J}$. However, as we've seen, the $\operatorname{div}$ of $\operatorname{curl}$ is not always zero, as verified by the spiral or helical field example. So, taking the $\operatorname{div}$ of equation $(3)$ and $(4)$ and using $(1)$ and $(2)$ yields a non-zero result.
$\mathbf{\nabla}\cdot(\mathbf{\nabla}\times\mathbf{D})=-\mathbf{\nabla}\cdot(\mathbf{\nabla}\times\mathbf{H})=\dfrac{\partial\rho}{\partial t}\tag8$
This is a beautiful expression of electrodynamics that unifies electricity and magnetism. It states that any flow of electromagnetic energy coupling with mechanical work is due to the current, placing it at the root. We can see from $(8)$, in generation of electricity by Faraday’s law, current increases and in mechanical work by Maxwell’s law current decreases. It is a force term that arises because the curl of electric and magnetic fields is higher near the axis and lower farther from it, thus forming a non-homogeneous field. These fields have sources and therefore lose energy, which can be transferred. To explain it further, let’s take an example of $\operatorname{curl}$ of $\mathbf{D}$ in Faraday’s law. The charge moves along the wire, and its path becomes helical as the conductor rotates. This helical motion causes the moment of $\mathbf{D}$ responsible for changing $\mathbf{H}$ to change, resulting in a gain of energy. The $\operatorname{curl}$ of $\mathbf{D}$ or $\mathbf{J}$ produces magnetic flux $\Phi$, but the rotation of the charge itself produces a moment of $\mathbf{D}$ that changes the magnetic field. Faraday's law is often misinterpreted as **change in magnetic field produces EMF **when, in fact, it's the $\mathbf{H}$ changed by the $\operatorname{curl}$ of $\mathbf{D}$. The traditional understanding of Faraday's law overlooks the crucial role of the moment of $\mathbf{D}$ in changing the magnetic field.
The divergence of the curl can indeed be non-zero, but this aspect has been obscured by the assumption that Maxwell's equations don't supported by moving charges in classical relativity. To address this, the theory of relativity was developed, which postulates that time is local and not absolute. This implies that the $\operatorname{curl}$ of $\mathbf{E}$ for a moving charge doesn't show any change in energy. The $\operatorname{div}$ of the $\operatorname{curl}$ of $\mathbf{E}$ is non-zero $(8)$ and equal to the $\operatorname{div}$ of the $\operatorname{curl}$ of $\mathbf{B}$. This leads to the extra $\mathbf{B}$ produced due to the motion of the charge in Faraday's law. From Lorentz transformation for relativistic motion and concept of local time, a relativistic correction term for $\mathbf{B}$ arises for moving charge. Because of this correction, the magnetic field need to modified as $\mathbf{B}\rightarrow\mathbf{B}-\mathbf{v}\times\mathbf{B}/c$ for a moving charge. But the correction term incorporated in Faraday’s law was $\mathbf{E}\rightarrow\mathbf{E}+\mathbf{v}\times\mathbf{E}/c$ for the effects of moving charge on the electric field. It was this correction that led to the development of the theory of relativity for mechanics, ensuring that the laws of electrodynamics remain invariant. So to envision that changing $\mathbf{E}$ produces changing $\mathbf{B}$ in closed loop once they get impulse from source and can moves too far without loss is an incorrect idea. So no such electromagnetic waves exist and no such motion as inertial for infinite length even in space of no matter.
$\quad\quad\ \mathbf{Appendix}$
Still, if someone wants the right-hand side of $(8)$ to equal zero and the improved equations to have the same properties as Maxwell's equations, then there is a correction term for $(3)$ and $(4)$ similar to what Maxwell had. Specifically, $\mathbf{H}\rightarrow\mathbf{H}+\mathbf{J}$ and $\mathbf{D}\rightarrow\mathbf{D}-\mathbf{J}/$.Then, our equations for Faraday's law and Maxwell's law are transformed as follows.
$\mathbf{\nabla}\times\mathbf{D}=\mathbf{H}+\mathbf{J}\tag9$
$\mathbf{\nabla}\times\mathbf{H}=\mathbf{D}-\mathbf{J}\tag{10}$
Now right-hand side of $(8)$ is zero. Using $(9)$ and $(10)$ in $(5)$ for deriving wave equations. We don’t have any wave equation like $(5)$ and $(6)$ with corrected $\mathbf{H}$and $\mathbf{D}$.
Let see how correction terms for $\mathbf{E}$ and $\mathbf{B}$ that changes $(3)$ and $(4)$ to $(9)$ and $(10)$, looks in relativistic form. We have from $(1)$ and $(2)$ for$\mu_0$ and $\epsilon_0$, $\mathbf{E}=-c^2\mathbf{B}$, using them for correction terms $\mathbf{E}-\mathbf{J}/\epsilon$ and $\mathbf{B}+\mu\mathbf{J}$. After some manipulation of putting $\mathbf{J}=\mu\mathbf{B}$ in $\mathbf{E}$ and $\mathbf{J}=\mathbf{E}/\epsilon$ in $\mathbf{B}$,
$\mathbf{E}\rightarrow\mathbf{E}+\dfrac{v^2}{c^2}\mathbf{E}\quad;\quad\mathbf{B}\rightarrow\mathbf{B}-\dfrac{v^2}{c^2}\mathbf{B}\tag*{}$
The relativistic correction term for the electric field in Faraday's law is $-v^2/c^2\,\mathbf{E}$, which leads to the expression $\mathbf{E}=\gamma^2\mathbf{E}$, where $\gamma$ is the Lorentz factor, $\gamma=1/\sqrt{1-v^2/c^2}$. Similarly, the relativistic correction term for the magnetic field in Ampere's law is also $-v^2/c^2\,\mathbf{B}$ which leads to the expression $\mathbf{B}=\mathbf{B}/\gamma^2$. The fact that both $\mathbf{E}$ and $\mathbf{B}$ receive the same correction term, $-v^2/c^2$, demonstrates the uniformity and symmetry between electricity and magnetism.
