Maxwell's equations are a set of [differential equations](Differential%20equations%20(index).md) that underpin the classical theory of Electromagnetism and govern all electromagnetic phenomena. The _macroscopic_ Maxwell's equations are given as 1. $\nabla\times \mathbf{H}(\mathbf{r},t)=\frac{\partial \mathbf{D}(\mathbf{r},t)}{\partial t}+\mathbf{J}(\mathbf{r},t)$ 2. $\nabla\times \mathbf{E}(\mathbf{r},t)=-\frac{\partial \mathbf{B}(\mathbf{r},t)}{\partial t}$ 3. $\nabla\cdot\mathbf{D}(\mathbf{r},t)=\rho(\mathbf{r},t)$ 4. $\nabla\cdot\mathbf{B}(\mathbf{r},t)=0$ where $\mathbf{H}(\mathbf{r},t)$ and $\mathbf{D}(\mathbf{r},t)$ are the [magnetic field](Magnetic%20field.md) and [electric displacement](Electric%20displacement.md) respectively, $\mathbf{E}(\mathbf{r},t)$ is the [Electric field](Electric%20field.md), $\mathbf{B}(\mathbf{r},t)$ is the [magnetic flux density](Magnetic%20flux%20density.md), $\mathbf{J}(\mathbf{r},t)$ is the [current density](Current%20density.md), and $\rho(\mathbf{r},t)$ is the [electric charge](Electric%20charge.md) density. In equations 1. and 2., the cross product with the [[Nabla]] is the [[Curl]], so we'll refer to these as _Maxwell's curl equations_ while the 2nd pair of equations, 3. and 4., we will refer to as _Maxwell's [[Divergence]] equations_. We refer to this system of differential equations as being the macroscopic Maxwell's equations since the fields described here are continuous. However, in reality matter contains only discrete charges, thus these equations describe averages over space rather than providing the full [microscopic](Maxwell's%20Equations%20in%20Quantum%20Mechanics.md) picture. %%What kind of differential equations are these?%% # Integral form of Maxwell's equations # Discussion ## In a vacuum The above [Maxwell's equations](Maxwell's%20equations.md) are generalized for all cases, while in a vacuum the first equation and third equation we wrote are rewritten as $\nabla\times\mathbf{B}(\mathbf{r},t)=\frac{1}{c^2}\frac{\partial\mathbf{E}(\mathbf{r},t)}{\partial t}+\mu_0\mathbf{J}(\mathbf{r},t)$ and $\nabla\cdot\mathbf{E}(\mathbf{r},t)=\frac{\rho}{\varepsilon_0}$ where we considered the fact that in a vacuum [electric displacement](Electric%20displacement.md#In%20a%20vacuum) and the vacuum [magnetic field](Magnetic%20field.md#In%20a%20vacuum) are $\mathbf{D}(\mathbf{r},t)=\varepsilon_0\mathbf{E}(\mathbf{r},t)$ and $\mathbf{H}(\mathbf{r},t)=\frac{1}{\mu_0}\mathbf{B}(\mathbf{r},t)$ respectively. Notice here that the [speed of light](The%20speed%20of%20light.md), $c$, appears here as well. This hints at the role of Maxwell's equations in modeling light propagation via the [[Electromagnetic wave equations]]. ## What each equation tells us ### Charge conservation [[Charge conservation]] is implied from [equation 1](Maxwell's%20equations.md) since $\nabla\cdot\nabla\times\mathbf{H}=0$ and substituting $\nabla\cdot\mathbf{D}$ gives the following continuity equation: $\nabla\cdot\mathbf{J}(\mathbf{r},t)+\frac{\partial \rho(\mathbf{r},t)}{\partial t}=0$ ## Derivation of the wave equations The [curl equations](Maxwell's%20equations.md) are used to derive the [electromagnetic wave equations](Electromagnetic%20wave%20equations.md), which are given as ## Properties ### Lorentz invariance # Derivations of Maxwell's equations ## Non-relativistic derivation ## Relativistic derivation Maxwell's equations may be formulated in [[Minkowski Space]] from the [electromagnetic field tensor](Electromagnetic%20field%20tensor.md). #Electromagnetism