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April 12, 2020
Author: Curtis Mobley
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Maxwell's Equations in Vacuo
This page begins a qualitative overview of Maxwell’s equations. Entire books have been written about these equations, so two pages are not going to teach you much. The goal here is to present the fundamental ideas and, hopefully, inspire you to continue to study these equations in the references provided. The discussion presumes a knowledge of basic physics (concepts such as electric charge and current, and electric and magnetic ﬁelds). Knowledge of vector calculus (divergence and curl in particular) is needed to understand the equations, but you can understand the basic ideas even without the math. If you are unfamiliar with the basic physics and math of electric and magnetic ﬁelds, or need a good review, an excellent place to start is A Student’s Guide to Maxwell’s Equations by Fleisch (2008). That tutorial spends 130 pages covering what is presented here.
Recall the Lorentz equation for the force $F$ exerted on an electric charge $q$ moving with velocity $v$ through an electric ﬁeld $E$ and a magnetic ﬁled $B$ (in SI units):
$$F=q\left(E+v\times B\right).$$ 
In this discussion, vectors in 3D space are indicated by boldfaced symbols. The Lorentz equation gives us the units for electric and magnetic ﬁelds. The force on the charge due to the electric ﬁled is $F=qE$, so the units of electric ﬁeld must be
$$\left[E\right]=\frac{\left[F\right]}{\left[q\right]}=\frac{newton}{coulomb}\phantom{\rule{0.3em}{0ex}},$$ 
where $\left[...\right]$ denotes ”units of ...”. Similarly, magnetic ﬁelds have units of
$$\left[B\right]=\frac{\left[F\right]}{\left[qv\right]}=\frac{newton}{coulomb\phantom{\rule{1em}{0ex}}meters\phantom{\rule{1em}{0ex}}per\phantom{\rule{1em}{0ex}}second}\phantom{\rule{0.3em}{0ex}}.$$ 
You will see equivalent forms for these units. A newton per coulomb is the same as a volt per meter. An ampere is a current of a coulomb per second, so we can write $\left[B\right]=N\u2215\left(A\phantom{\rule{0.3em}{0ex}}m\right)$, which is called a Tesla (T). Table 1 summaries for reference the quantities seen in Maxwell’s equations.
By the way, an electric ﬁeld of 1 V/m is a very weak ﬁeld: just think of a large parallel plate capacitor with the plates separated by 1 m and connected by a 1 V battery. The electric ﬁeld between a thundercloud and the ground is of order $1{0}^{5}\phantom{\rule{1em}{0ex}}V\u2215m$ just before a lightning discharge. On the other hand, a 1 T magnetic ﬁeld is really strong. The Earth’s magnetic ﬁeld at the surface is about $5\times 1{0}^{5}\phantom{\rule{1em}{0ex}}T$. Important research has shown that a 16 T magnetic ﬁeld is so strong that it can overcome the force of gravity and levitate a living frog (Berry and Geim, 1997. Eur. J. Phys 18, 307313).
Physical  Symbol  SI Units  Comment 
Permittivity  ${\mathit{\epsilon}}_{o}$  $=8.85\times 1{0}^{12}{A}^{2}\phantom{\rule{0.3em}{0ex}}{s}^{4}\phantom{\rule{0.3em}{0ex}}k{g}^{1}\phantom{\rule{0.3em}{0ex}}{m}^{3}$


Permeability  ${\mu}_{o}$  $=4\pi \times 1{0}^{7}kg\phantom{\rule{0.3em}{0ex}}m\phantom{\rule{0.3em}{0ex}}{s}^{2}\phantom{\rule{0.3em}{0ex}}{A}^{2}$


Electric charge  q  coulomb (C) 

Charge density  $\rho $  $C\phantom{\rule{0.3em}{0ex}}{m}^{3}$  charge per unit volume 
Electric current  $I$  ampere (A)  SI base unit 
Current density  $J$  $A\phantom{\rule{0.3em}{0ex}}{m}^{2}$  current per unit area 
Electric ﬁeld  $E$  $N\u2215C=V\u2215m$ 

Magnetic ﬁeld  $B$  $N\u2215\left(A\phantom{\rule{0.3em}{0ex}}m\right)=T$ 

Electric dipole  $p$  $C\phantom{\rule{0.3em}{0ex}}m$ 

Polarization  $P$  $C\phantom{\rule{0.3em}{0ex}}m\u2215{m}^{3}$  electric dipole moment per unit volume 
Magnetic dipole  $m$  $A\phantom{\rule{0.3em}{0ex}}{m}^{2}$ 

Magnetization  $M$  $\left(A\phantom{\rule{0.3em}{0ex}}{m}^{2}\right)\u2215{m}^{3}$  magnetic dipole moment per unit volume 
Electric  $D$  $C\u2215{m}^{2}$  $D={\mathit{\epsilon}}_{o}E+P$ 
Magnetic  $H$  $A\u2215m$  $H=B\u2215{\mu}_{o}M$ 
Maxwell’s Equations in Vacuo
Without further ado, Maxwell’s equations for the electric ﬁeld $E\left(x,t\right)$ and magnetic ﬁeld $B\left(x,t\right)$ in a vacuum are (in diﬀerential form, in SI units)
$$\begin{array}{lll}\hfill \nabla \cdot E=& \phantom{\rule{1em}{0ex}}\frac{1}{{\mathit{\epsilon}}_{o}}\rho \phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \hfill \nabla \cdot B=& \phantom{\rule{1em}{0ex}}0\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \nabla \times E=& \phantom{\rule{1em}{0ex}}\frac{\partial B}{\partial t}\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \nabla \times B=& \phantom{\rule{1em}{0ex}}{\mu}_{o}J+{\mu}_{o}{\mathit{\epsilon}}_{o}\frac{\partial E}{\partial t}\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\end{array}$$(If you are not familiar with the divergence $\nabla \cdot V$ and the curl $\nabla \times V$ of a vector ﬁeld $V$, these are combinations of the spatial derivatives of the components of V. They are deﬁned in the box below .) Note that ”in a vacuum” means that the electric and magnetic ﬁelds are in empty space. There can still be electric charges located here and there in space (the $\rho $ term), and the same for currents ($J$), which give rise to the ﬁelds in the region of interest.
These equations can be described as follows:
 Eq.(1)
 This equation is called Gauss’s law for electric ﬁelds. It shows how electric charges (the charge density $\rho $) create electric ﬁelds. This equation is the equivalent of Coulomb’s law for a point charge.
 Eq.(2)
 This equation is sometimes called Gauss’s law for magnetic ﬁelds. It says that there are no magnetic charges corresponding to electric charges.
 Eq.(3)
 This is Faraday’s law. It shows that a timevarying magnetic ﬁeld creates an electric ﬁeld.
 Eq.(4)
 This is Ampere’s law as modiﬁed by Maxwell. The ﬁrst term on the right, deduced by Ampere, shows that electric currents create magnetic ﬁelds. The second term on the right, added by Maxwell, shows that a timevarying electric ﬁeld also creates a magnetic ﬁeld.
Thus there are two ways to create electric ﬁelds: electric charges create them, and timedependent magnetic ﬁelds create them. One might suppose that the electric ﬁelds resulting from these two entirely diﬀerent creation mechanisms could some way be diﬀerent, but they are not. An electric ﬁeld is an electric ﬁeld, no matter how it is created. That’s just the way the universe works. (Pondering this equivalence of electric ﬁelds, no matter how created, was one of the things that lead Einstein to the development of special relativity.) The same situation holds for magnetic ﬁelds. They can be created by electric currents or by timedependent electric ﬁelds, but the nature of the magnetic ﬁeld is the same in either case.
Simply stating Maxwell’s equations is really no diﬀerent than simply stating Newton’s law of gravity for the magnitude of the force of attraction between two spherical masses ${M}_{1}$ and ${M}_{2}$ separated by a distance $r$:
$$F=G\frac{{M}_{1}{M}_{2}}{{r}^{2}}\phantom{\rule{0.3em}{0ex}}.$$  (5) 
Newton didn’t derive his law of gravity from more fundamental principles; it is the fundamental principle. Newton found that if he assumed Eq. (5) to be true, then he could derive Kepler’s laws of planetary motion, the motion of the moon, and (to ﬁrst order) the ocean tides. The same can be said of Maxwell’s equations. They are based on decades of observational work by Coulomb, Gauss, Faraday, Ampere and others, but we can view them as the mathematical statement of the fundamental laws governing electric and magnetic ﬁelds. We can simply accept these equations as given and get on with the business of applying them to problems of interest. (Of course, ”fundamental laws of nature” may turn out of be imperfect in the light of new data. That happened to Newton’s law of gravity, which was replaced by, and can be derived from, Einstein’s theory of general relativity. Likewise, Maxwell’s equations can now be derived from the more fundamental laws of quantum electrodynamics developed by Feynman and others.)
It may at ﬁrst glance seem that Maxwell’s equations are overdetermined. That is, there are four equations but only two unknowns, $E$ and $B$. This would be true for algebraic equations, in which case we could solve two linearly independent equations for two unknowns. However, for vector ﬁelds, Helmholtz’s theorem (also known as ”the fundamental theorem of vector calculus”) says that an arbitrary vector ﬁeld in 3 dimensions can be uniquely decomposed into a divergence part (with zero curl) and a curl part (with zero divergence) (under a few conditions, namely vector functions that are suﬃciently smooth and that decay to zero at inﬁnity). Conversely, knowing the divergence and curl of a vector ﬁeld determines the vector ﬁeld. That is the case here for both $E$ and $B$. Given the charge density $\rho $ and current density $J$, the four Maxwell equations uniquely determine the electric and magnetic ﬁelds via their divergences and curls. (To be rigorous, a vector ﬁeld is determined from its divergence and curl to within an additive term. This is somewhat like saying that knowing a derivative $df\left(x\right)\u2215dx$ determines $f$ to within an additive constant. Adding a boundary condition $f\left({x}_{o}\right)={f}_{o}$ then ﬁxes the value of the constant.)
Light as an Electromagnetic Phenomenon
Starting with equations (1) to (4), Maxwell derived what is probably the most elegant and important result in the history of physics. Consider a region of space where there are no charges ($\rho =0$) or currents ($J=0$). Equations (1)(4) then become
$$\begin{array}{lll}\hfill \nabla \cdot E=& \phantom{\rule{1em}{0ex}}0\phantom{\rule{2em}{0ex}}& \hfill \text{(6)}\\ \hfill \nabla \cdot B=& \phantom{\rule{1em}{0ex}}0\phantom{\rule{2em}{0ex}}& \hfill \text{(7)}\\ \hfill \nabla \times E=& \phantom{\rule{1em}{0ex}}\frac{\partial B}{\partial t}\phantom{\rule{2em}{0ex}}& \hfill \text{(8)}\\ \hfill \nabla \times B=& \phantom{\rule{1em}{0ex}}{\mu}_{o}{\mathit{\epsilon}}_{o}\frac{\partial E}{\partial t}\phantom{\rule{2em}{0ex}}& \hfill \text{(9)}\end{array}$$Now take the curl of Eq. (8), use the vector calculus identity $\nabla \times \left(\nabla \times E\right)=\nabla \left(\nabla \cdot E\right){\nabla}^{2}E$, use Eq. (6) to eliminate the $\nabla \left(\nabla \cdot E\right)$ term, and use Eq. (9) to rewrite the $\partial \left(\nabla \times B\right)\u2215\partial t$ term. The result is
$${\nabla}^{2}E={\mu}_{o}{\mathit{\epsilon}}_{o}\frac{{\partial}^{2}E}{\partial {t}^{2}}\phantom{\rule{0.3em}{0ex}}.$$ 
The same process starting with the curl of Eq. (9) gives an equation of the same form for $B$. Equations of the form
$${\nabla}^{2}f=\frac{{\partial}^{2}f}{\partial {x}^{2}}+\frac{{\partial}^{2}f}{\partial {y}^{2}}+\frac{{\partial}^{2}f}{\partial {z}^{2}}=\frac{1}{{v}^{2}}\frac{{\partial}^{2}f}{\partial {t}^{2}}$$ 
describe a wave propagating with speed $v$. Thus each component of $E$ and $B$ satisﬁes a wave equation with a speed of propagation
$$v=\frac{1}{\sqrt{{\mu}_{o}{\mathit{\epsilon}}_{o}}}\phantom{\rule{0.3em}{0ex}}.$$  (10) 
Inserting the experimentally determined values of ${\mu}_{o}$ and ${\mathit{\epsilon}}_{o}$ given in Table 1 gives $v=3\times 1{0}^{8}\phantom{\rule{1em}{0ex}}m\phantom{\rule{0.3em}{0ex}}{s}^{1}$. As Maxwell observed (in A Dynamical Theory of the Electromagnetic Field, 1864, §20), ”This velocity is so nearly that of light that it seems we have strong reason to conclude that light itself (including radiant heat and other radiations) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic ﬁeld according to electromagnetic laws.” This conclusion is one of the greatest intellectual achievements of all time: not only were electric and magnetic ﬁelds tied together in Maxwell’s equations, but light itself was shown to be an electromagnetic phenomenon. This is the ﬁrst example of a ”uniﬁed ﬁeld theory,” in which seeming independent phenomena—here electric ﬁelds, magnetic ﬁelds, and light—were shown to related and governed by the same underlying equations.