# Conceptual Understanding in Introductory Physics XVIII

The Maxwell equations contain everything mentioned in an introductory calculus-based electromagnetic theory course, and then some. They contain detailed mathematical structure and deep insight into electromagnetic fields. They are a magnificent playground for learning the various theorems of vector calculus, and applying these theorems to the fields of particles naturally leads to things like divergence, curl, flux, circulation, and the Dirac delta function, something no introductory course I know of incorporates (Matter & Interactions mentions it, as it does the differential vector operators). The Maxwell equations also make a great starting point for alternative ways of thinking about vector fields, like geometric algebra, differential forms and tensors. Of course the culmination of any introductory electromagnetic theory course is “solving for light” or more correctly, solving the Maxwell equations in free space and showing how the solution leads to light as a pulse of electromagnetic energy. If you start your mechanics course or electromagnetic theory course with special relativity, you can come full circle and demonstrate how the Maxwell equations are relativistically rigorous under a Lorentz transformation, bringing students back to the topic that prompted Einstein to think about relativity in the first place.

I’m saving the best motivation for last though. The Maxwell equations are the best place in introductory physics to let students practice with symmetry and duality. Mere visual inspection of the equations, especially when written for free space, leads to adding terms that make the equations fully symmetric under a duality transformation (see here and here) and predicting the existence of magnetic monopoles. As one of my students said just last week, “The symmetry of Maxwell’s equations is so beautiful that magnetic monopoles just have to exist!” I don’t know that that argument will hold up to current experimental tests, but it’s great to see students understand how symmetry leads to predictions that lead to experiments. That’s meaningful.

So here are some questions on the Maxwell equations, some of which I’ve used at the end of second semester calculus-based physics.

(a) Write Maxwell’s equations in integral form. Notation must be consistent and correct.

(b) For each integral equation, write a one sentence interpretation of that equation. Do not simply write a verbal translation of the equation; interpret the equation’s meaning. There is probably more than one correct interpretation for each equation.

(c) Transform each equation into a differential equation, showing each step clearly. Notation must be consistent and correct.

(d) For each differential equation, write a one sentence interpretation of that equation. Do not simply write a verbal translation of the equation; interpret the equation’s meaning. Again, there is probably more than one correct interpretation for each equation.

(e) For each differential or integral equation, give one specific application of that equation from this course. Do not cite trivial applications.

(f) Using symmetry arguments, rewrite each set of equations (the integral set and the differential set) to account for the existence of magnetic monopoles.

(g) Write each set of equations for free space. Comment on how the free space equations would look if magnetic monopoles are found.

These questions themselves could conceivably serve as an entire final exam. Give it a try!

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