Electrodynamics/Maxwell's Equations

Maxwells Equations

When James Clarke Maxwell was doing his work with electrodynamics, several of the concepts that we have been considering had not yet been introduced to the world of mathematics. For instance, vector calculus was a very young discipline, and many of the operators currently in use (Div, Curl, the Laplacian) did not exist in Maxwell's time.

The original "Maxwells Equations" were a set of 20 complicated differential equations that placed a primary focus on the idea of magnetic potential (a quantity which is almost completely ignored in the modern variants of these equations).

Heinrich Hertz and Oliver Heaviside did much of the work to convert Maxwells original equations into a more convenient form. The Electric and Magnetic fields were deemed to be of primary importance, whereas the magnetic potential was dropped from the formalization. From Hertz and Heaviside we obtained the 4 equations that we know today as "Maxwell's Equations".

The 4 Equations

Here are Maxwell's equations. Several of these equations have been seen already in previous chapters.

[Gauss' Law of Electrostatics]

$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$

[Gauss' Law of Magnetostatics]

$\nabla \cdot \mathbf {B} =0$

[Faraday's Law]

$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$

[Ampere-Maxwell Law]

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$

Where: $\rho$ is the charge density, which can (and often does) depend on time and position, $\epsilon _{0}$ is the permittivity of free space, $\mu _{0}$ is the permeability of free space, and $\mathbf {J}$ is the current density vector, also a function of time and position. The units used above are the standard SI units. Inside a linear material, Maxwell's equations change by switching the permeability and permitivity of free space with the permeability and permitivity of the linear material in question. Inside other materials which possess more complex responses to electromagnetic fields, these terms are often represented by complex numbers, or tensors.

We can write Maxwell's equations in another form, that relates each field to its sources: By taking the curl of the third equation, we get

$\nabla \times \nabla \times \mathbf {E} =-{\frac {\partial }{\partial t}}\nabla \times \mathbf {B}$

Since

$\nabla \times \nabla \times \mathbf {E} =\nabla (\nabla \cdot \mathbf {E} )-\nabla ^{2}\mathbf {E} ={\frac {1}{\epsilon _{0}}}\nabla \rho -\nabla ^{2}\mathbf {E}$

and since

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$

we get

${\frac {1}{\epsilon _{0}}}\nabla \rho -\nabla ^{2}\mathbf {E} =-{\frac {\partial }{\partial t}}(\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}})$

or $\nabla ^{2}\mathbf {E} -\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=\mu _{0}{\frac {\partial \mathbf {J} }{\partial t}}+{\frac {1}{\epsilon _{0}}}\nabla \rho$

Similarly, $\nabla ^{2}\mathbf {B} -\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}=\mu _{0}\nabla \times \mathbf {j}$

Using Tensors

It should be noticed, if not immediately, that the first two equations are essentially equivalent, and that the second two equations have a similar form and should be able to be put into a single form. We can use our field tensors F and G to put the 4 Maxwell's equations into two more concise equations:

$\sum _{\mu }{\frac {\partial \mathbb {F} _{\mu \nu }}{\partial x_{\mu }}}=4\pi j_{\nu }$

$\sum _{\mu }{\frac {\partial \mathbb {G} _{\mu \nu }}{\partial x_{\mu }}}=0$

Symmetry

You may notice that these two equations are very similar, but they are not completely symmetric. The magnetic field equations reduce because magnetic fields always have two opposite poles, whereas an electric field may have only a single charge. This lack of symmetry in these equations has prompted scientists to search for a magnetic monopole, something that we will talk about in later chapters.

Besides the forms of these equations, modern "unified theories" of physics seeking to describe all forces of nature (including, significantly, electromagnetism and gravity) often posit the existence of monopoles. As a basic consideration, similarity and symmetry among many equations and processes in physics often leads to the discovery of entirely new entities or phenomena. Thus, the pronounced lack of symmetry between the magnetic and electric field equations is a simple and logical reason for scientists to search for monopoles.