The Maxwell Equations

Electromagnetic waves are the carrier of all target relevant information between a given radar and a distant target observed by that radar.
Michael Faraday (1791-1867) discovered, that a time varying magnetic flux $\vec{B}(\vec{r},t)$ through a closed conducting loop induces an electromagnetic field around that loop.

$$\oint_C\vec{E}(\vec{r},t)\cdot dl = - \iint_A \frac{\partial \vec{B}(\vec{r},t)}{\partial t}\cdot d \vec{S}$$ (A.1)

where $\vec{E}(\vec{r},t)$ is the electric field intensity vector, C a closed path around $\vec{B}(\vec{r},t)$ and A the open area bounded by C. This rule is known as Faraday's Induction Law, stating that a time varying magnetic field is always connected to an electric field.
The second important equation, known as the Ampère 's Circuital Law, was found by André Marie Ampère (1775-1836). This equation relates a line integral of $\vec{B}(\vec{r},t)$ tangent to a closed curve C, with the total current i passing within the confines of C.

$$\oint_C\vec{B}(\vec{r},t)\cdot d \vec{l} = \mu \iint_A\vec{J_i}(\vec{r},t) \cdot d \vec{S} = \mu i$$ (A.2)

where $\vec{J_i}(\vec{r},t)$ is the induced current density and $\mu$ is the relative permeability tensor for the given media. Even though the equation above is valid for many applications it does not yield the whole truth. Moving charges are not the only source of a magnetic field, e.g. the for the case of charging a capacitor a B-field can be measured between the plates, even though no current traverses the capacitor. James Clerk Maxwell (1831-1879), therefore, predicted the existence of a another type of current-flow mechanism, called displacement current density $\vec{J_e}(\vec{r},t)$

$$\vec{J_e}(\vec{r},t) = \varepsilon \frac{\partial \vec{E}(\vec{r},t)}{\partial t}$$ (A.3)

where $\varepsilon$ is the dielectric constant leading to a new and complete formulation of (A.2)

$$\oint_C\vec{B}(\vec{r},t)\cdot d \vec{l} = \mu \iint_A {\left(\ve... ...arepsilon\frac{\partial \vec{E}(\vec{r},t)}{\partial t}\right)} \cdot d \vec{S}$$ (A.4)

meaning that a time varying E-field is always accompanied by an magnetic B-field, even when $\vec{J_i}(\vec{r},t) = 0$. The remaining two Maxwell equations refer to Karl Friedrich Gauss (1777-1855). The first one denotes the relation between the electric field $\vec{E}(\vec{r},t)$ through a closed surface A and the charges contained in the volume V surrounded by A.

$$\oiint_A\vec{E}(\vec{r},t)\cdot d \vec{S} = \frac{1}{\epsilon} \iiint_V \rho dV$$ (A.5)

where $\rho$ is the electric charge density and $d \vec{S}$ is a vector perpendicular to A pointing outwards. This equation is sometimes referred as the coulomb law.
Due to the fact, that no magnetic monopoles are known to exist, the equivalent of the equation above for the magnetic flux $\vec{B}$ is given by

$$\oiint_A\vec{B}(\vec{r},t)\cdot d \vec{S} = 0$$ (A.6)

The set of the four integral equations are know as the Maxwell Equations and describe the behavior and relation of electromagnetic fields. These equations can be also written in differential formulation which is better suited to derive the wave aspects of the electromagnetic field. In order to change from the integral to the differential formulation Gausses theorem (1813)

$$\oiint_A \vec{X} \cdot d \vec{S} = \iiint_V \nabla \cdot \vec{X} dV$$ (A.7)

and Stokes' theorem (1854)

$$\oiint_C \vec{X} \cdot d \vec{S} = \iint_A (\nabla \times \vec{X}) \cdot d \vec{S}$$ (A.8)

are applied, where $\vec{X}$ denotes a vector field. By applying Stokes' theorem to the electric field we get

$$\oiint_C \vec{E}(\vec{r},t)\cdot d \vec{S} =\iint_A \nabla \times\vec{E}(\vec{r},t)\cdot d \vec{S}$$ (A.9)

Comparing this result with (A.1) we find

$$\iint_A \nabla \times\vec{E}(\vec{r},t)\cdot d \vec{S} = - \iint_A \frac{\partial \vec{B} (\vec{r},t)}{\partial t}\cdot d \vec{S}$$ (A.10)

This must be true for all surfaces A confined by C, which is only the case if the arguments under the integrals are equal, hence

$$\nabla \times \vec{E}(\vec{r},t)= - \frac{\partial \vec{B}(\vec{r},t)}{\partial t}$$ (A.11)

In exactly the same way (A.4) is translated into

$$\nabla \times\vec{B}(\vec{r},t)= \mu \left(\vec{J}_{i}(\vec{r},t)+\varepsilon\frac{\partial \vec{E}(\vec{r},t) }{\partial t}\right)$$ (A.12)

Gausses theorem applied to the electric field yields

$$\oiint_A\vec{E}(\vec{r},t)\cdot d \vec{S} = \iiint_V \nabla \cdot\vec{E}(\vec{r},t)dV$$ (A.13)

and with (A.5) we get

$$\iiint_V \nabla \cdot\vec{E}(\vec{r},t)dV = \frac{1}{\epsilon} \iiint_V \rho dV$$ (A.14)

which in the differential form yields

$$\nabla \cdot\vec{E}(\vec{r},t)= \frac{\rho}{\epsilon}$$ (A.15)

Analogous application of Gausses theorem to (A.5) leads to

$$\nabla \cdot\vec{B}(\vec{r},t)= 0$$ (A.16)

With these equations, which are valid for all times t and all points in space $\vec{r}$, the behavior of electromagnetic fields is unambiguously described by the real quantities $\vec{E}(\vec{r},t)$, $\vec{B}(\vec{r},t)$, $\vec{J}(\vec{r},t)$ and $\rho(\vec{r},t)$.