Maxwell's Equations Trivia Quiz

Answer: A quantity that has both size and a direction in space

Physicists and mathematicians find it useful to make distinctions between two types of quantities that you can measure. One is a scalar, or an ordinary number. It can be used to describe how much something weighs, or how much time is left until supper, or the distance between two cities. The other is a vector, which has both a size (or magnitude) and a direction. Famous vectors include velocity (how fast is that car moving, and in what direction?), force (you don't understand Earth's gravity if you don't know that it's pointing down), and even position (which gives you both distance and direction from a starting point).

Electric and magnetic field vectors have size and direction. The direction gives you a sense of which way a charged particle would move in such a field; it moves along the electric field lines (or opposite to them) if it has a positive (or negative) charge, and perpendicular to the magnetic field lines. The size gives you a sense of how much force is applied to a given charge. (An electric field applies a force equal to its own magnitude, times the size of the charge; the force applied by a magnetic field also depends on the charge's velocity.) As you can see, both magnitude and direction are crucial to understanding how the fields operate!

Answer: The degree to which the vectors spread out from a point

The classic illustration of divergence is a point surrounded by a ring of arrows; arrows, since they have both size and direction, are the traditional way to represent vectors. If the arrows point outward, away from the center, then the point is a "source" (like a faucet) and the field has positive divergence there; if they point inward, then the field has negative divergence and the point is a "sink" (or a drain).

Not all divergence calculations are quite so straightforward to draw, but the basic idea is the same: divergence measures the degree to which a particular point or region is "special," so that vectors spread out from it or head into it.

Answer: The electric-field strength passing through the surface

Here's how to take the surface integral of an electric field: first, take a surface, any surface. It could be closed, like the surface of the Earth, or it could be open, like the pleated surface of a fan. Next, divide the surface up into infinitesimally small pieces; these are called area elements, and are usually represented by dA, where "d" is a way of marking what you're integrating over, and A is a vector that's at a right angle to the surface. For each area element, take the dot product of the electric field with A, which picks out the part of the electric field that's perpendicular to that part of the surface. Now add up the result for all the area elements, and you're done! What you've got is a "flow" of electric field per area, a measure of just how much electric-field strength is passing through your surface.

Of course, this is literally going to take forever if the area elements are really infinitesimal, but luckily you can use calculus to get the answer more quickly. A bigger problem is how you know whether your result is positive or negative: there are two ways in which A could be pointing, and they have different signs! For a closed surface, convention says that outward is positive; for an open surface, you just have to pick a direction and make sure to keep track of it.

You can compute the flux of a magnetic field - or any vector field! - in the same way.

Answer: It's proportional to the charge inside

These are really just two equivalent ways of stating Gauss's law. Start with the differential form of the law: the divergence of the electric field E is equal to the charge density, rho, times a constant. Now let's integrate both sides over some volume. (Integration is a mathematical method for adding up all the values of an expression in some region, in this case a volume of space.) A charge density has units of charge per volume, so when you integrate it over a volume, you end up with the total charge. Meanwhile, there's a theorem of calculus that tells us how to integrate the divergence of E over a volume: it's the same as integrating E over the surface that encloses the volume, and we know that's the definition of the flux. So, there we have it: the flux of the electric field through a closed surface (that is, a surface that goes all the way around a volume) is equal to the charge inside, times a constant.

What's the constant? This depends on what units you're using and what material you're studying. In a vacuum and in SI units, which include such favorites as meters, Volts, and Amperes, the constant of proportionality here is one divided by epsilon0, the permittivity of free space. This has been measured to be 8.85 million-millionths Coulomb-squared per Newton per meter-squared.

Answer: Magnetic charges (i.e. monopoles) do not exist.

A magnetic monopole would be a net magnetic charge, similar to an electric charge -- but magnets don't seem to work that way. A permanent magnet always has a north pole and a south pole: no net magnetic charge. Electromagnets work the same way. Yet, if magnetic monopoles existed, they might illuminate a great deal -- for example, Paul Dirac showed that their existence would explain why electric charges come in discrete units, or quanta. String theory, and attempts at Grand Unified Theories (combinations of electromagnetic, weak and strong forces), all require monopoles -- so, of all four of Maxwell's equations, this is the one that inspires substantial experimental effort to disprove it.

Some experiments use sensitive magnets to look for natural monopoles; others look for evidence that they are being created in high-energy collisions in particle accelerators. Perhaps there will be exciting monopole news in the next few years!

Answer: The degree to which the vectors curl around a point

Consider a sink filled with water. At every point, the water is moving in a certain direction at a certain speed: we can describe its velocity with a vector function. Once we pull the plug, the water flows toward the drain, circling around it in a whirlpool: the velocity vectors "curl" around the drain. (It's a curious fact that plumbing metaphors are tremendously useful not only in vector calculus, but also in electrodynamics as a whole.) The degree and direction of a vector field's "curl" around a given point are represented by a vector quantity known (appropriately enough) as the curl of the vector field.

Answer: The magnetic field, B

The curl of the magnetic field is equal to mu0 times the current density, in a steady state and in SI units. If you integrate both sides around a loop -- like a loop of wire, for example -- then you find that the total magnetic field around the loop is mu0 times the total current that passes through the surface that the loop encloses. It's the magnetic equivalent of Gauss's law. Currents -- which are really just electrical charges in motion -- give rise to magnetic fields.

By the way, the constant mu0 is defined exactly rather than measured experimentally, since it's used to define units. It's exactly equal to 4 pi times 10^(-7) Newtons per Ampere-squared.

Answer: A changing magnetic field induces an electric field

You can find the rate of change of the magnetic field by taking its first derivative with respect to time -- another way in which calculus makes physics easier. Faraday formulated his law after a series of experiments with an electromagnet and a loop of wire. David J. Griffiths, in his "Introduction to Electrodynamics," has a concise summary of the experiments, which I'll borrow here:

"Experiment 1. He pulled a loop of wire to the right through a magnetic field.

"Experiment 2. He moved the magnet to the left, holding the loop still.

"Experiment 3. With both the loop and the magnet at rest, he changed the strength of the field."

In each case, a current flowed around the loop. (This is how generators work!)

As before, we can get a new perspective on this equation by integrating both sides. We then see that the integral of the electric field around a closed loop is equal to -1 times the rate of change in magnetic flux through the loop. That -1 just indicates that the flux from the induced electric field is in the opposite direction from the change in the magnetic flux.

Answer: The derivative of the electric field with respect to time

The corrected Ampère's law is the fourth and last of Maxwell's equations: in a vacuum, the curl of the magnetic field is equal to mu0 times the current density plus mu0 times epsilon0 times the derivative of the electric field with respect to time. (It's much more concise and beautiful written out mathematically, but that takes extra symbols and Greek letters). The new equation has a striking symmetry with Faraday's law: now we see that a changing electric field induces a magnetic field, just as a changing magnetic field induces an electric field.

Maxwell didn't derive, prove, or measure any of the four Maxwell's equations, but he did unite them into one coherent framework. The phrase "displacement current" is just about the only thing about his theory that isn't beautiful: the only thing that's current-like about the term is its units!

Answer: The speed of light

For the electric field, you get this equation: the Laplacian of E equals mu0 times epsilon0 times the second derivative of E with respect to time. (That second derivative gives the rate of change of the rate of change: it would be zero if E were changing at a constant rate. The wave equation for the magnetic field looks exactly the same, but with B substituted for E.) That's just like the equation for a three-dimensional wave moving at a speed of one over the square root of mu0 times epsilon0 -- and, as it happens, that value is exactly the speed of light in a vacuum.

Maxwell was dumbfounded by this result and wrote, "We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena." That medium, we now know, does not exist -- but light is no less than an electromagnetic wave, carrying an electric field and a magnetic field perpendicular to each other and to its direction of motion.

It's also a particle, of course. But that's a story for another time.