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.
Physical Preliminaries: Electric and Magnetic Fields
Recall the Lorentz equation for the force exerted on an electric charge moving with velocity through an electric ﬁeld and a magnetic ﬁled (in SI units):
In this discussion, vectors in 3D space are indicated by bold-faced symbols. The indicates the vector cross product. The Lorentz equation gives us the units for electric and magnetic ﬁelds. The force on the charge due to the electric ﬁled is , so the units of electric ﬁeld must be
where denotes ”units of ...”. Similarly, magnetic ﬁelds have units of
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 , which is called a Tesla (T). Table 1 summaries for reference the quantities seen in Maxwell’s equations.
The ﬁrst two quantities in Table 1 are worthy of comment. The electric constant or permittivity of free space, , is an empirical constant that measures an electric ﬁeld’s ability to “penetrate” a vacuum. In other words, it sets the strength of the force between two electric charges separated by some distance in a vacuum. This is seen if you write Coulomb’s law as
where is the magnitude of the force (in newtons) between charges and (in coulombs) separated by a distance (in meters) in a vacuum. The value of is not derived from fundamental physics; it must be measured. This can be done by measuring the force between two charges, but is more accurately measured with a parallel plate capacitor. Similarly, the magnetic constant or permeability of free space, , measures a magnetic ﬁeld’s ability to penetrate a vacuum. It sets the strength of the magnetic force between two current-carrying wires separated by some distance in a vacuum. It also must be measured. Why do these two fundamental constants have the particular values shown in Table 1? This is a question like “why does an electron have the charge it has, and not some other value?” All that can be said is that these values are what they are because that is just how the universe works.
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 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 . 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, 307-313).
a vacuum’s ability to
support an electric ﬁeld;
a vacuum’s ability to
support a magnetic ﬁeld;
a fundamental physical quantity
charge per unit volume
ampere (A = C/s)
measures ﬂow of electric charge per unit time
current per unit area
ﬁeld set up by stationary
a vector ﬁeld set up by
moving electric charges
electric dipole moment per unit volume
measures the magnetic
ﬁeld set up by a loop of
magnetic dipole moment per unit volume
Mathematical Preliminaries: Divergence and Curl
In order to enjoy Maxwell’s equations, it is necessary to understand the mathematical notation. For the beneﬁt of readers who are not familiar with vector calculus, the needed operations are as follows.
A scalar ﬁeld associates a number with each point in space and time. An example is the temperature in room. A vector ﬁeld associates a vector (a magnitude and a direction)
with each point in space and time. An example is the wind blowing outside your home.
The “del” operator (sometimes also called “nabla”) can be thought of as a vector whose elements are partial derivative operators deﬁned (in cartesian coordinates) as
Applying the del operator to a scalar gives a vector, called the gradient of the scalar ﬁeld:
Just like any vector, we can take the dot product of with a vector, and the result is a scalar. Taking the dot product of the del operator with a gradient gives a scalar:
This is usually called the Laplacian of , and is called the Laplace operator.
The divergence of a vector ﬁeld is deﬁned as
The cross product of two vectors and is
In the same fashion we get the curl of a vector ﬁeld, which is the cross product of with the vector ﬁeld and yields a vector:
There is a useful trick for remembering the order of the vector components and derivatives in the curl if you know how to expand the determinant of a matrix. Write the unit direction vectors in the ﬁrst row of the determinant, the partial derivatives in the second row, and the vector components in the third row:
Then expand the determinant just as though the elements were ordinary numbers, and let the derivatives operate on the vector elements.
Thus the divergence and curl are just certain combinations of the spatial derivatives of a vector ﬁeld. Each has a physical interpretation when the vector ﬁeld is a physical variable such as the velocity or an electric ﬁeld. However, just knowing the deﬁnitions is suﬃcient for our level of presentation of Maxwell’s equations.
Maxwell’s Equations in Vacuo
Without further ado, Maxwell’s equations for the electric ﬁeld and magnetic ﬁeld in a vacuum are (in diﬀerential form, in SI units)
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 term), and the same for currents (), which give rise to the ﬁelds in the region of interest.
These equations can be described as follows:
- This equation is called Gauss’s law for electric ﬁelds. It shows how electric charges (the charge density ) create electric ﬁelds. This equation is the equivalent of Coulomb’s law for a point charge.
- This equation is sometimes called Gauss’s law for magnetic ﬁelds. It says that there are no magnetic charges corresponding to electric charges.
- This is Faraday’s law. It shows that a time-varying magnetic ﬁeld creates an electric ﬁeld.
- 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 time-varying electric ﬁeld also creates a magnetic ﬁeld.
Thus there are two ways to create electric ﬁelds: electric charges create them, and time-dependent 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 time-dependent 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 and separated by a distance :
Newton did not 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 over-determined. That is, there are four equations but only two unknowns, and . 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 and . Given the charge density and current density , 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 determines to within an additive constant. Adding a boundary condition 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 () or currents (). Equations (1)-(4) then become
Now take the curl of Eq. (8), use the vector calculus identity , use Eq. (6) to eliminate the term, and use Eq. (9) to rewrite the term. The result is
The same process starting with the curl of Eq. (9) gives an equation of the same form for . Equations of the form
describe a wave propagating with speed . Thus each component of and satisﬁes a wave equation with a speed of propagation
Inserting the experimentally determined values of and given in Table 1 gives . 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.