1. The problem statement, all variables and given/known data This is a general question that applies to many homework problems (and real world problems), but I will provide an example to help guide the discussion. I am hoping you all can give me some examples of particularly clever manipulations of Maxwell's equations to make a difficult problem easier. For example, Griffiths' E&M (4th Edition) problem 7.22b states (paraphrased): A current I flows in a small circular loop of wire of radius a. This loop of wire sits above another loop of wire of radius b (where b > a). They are coaxial (the planes they span are parallel), and are separated by a distance z. Find the flux through the big loop (of radius b). 2. Relevant equations Φ = ∫B⋅da 3. The attempt at a solution The magnetic field of the top loop can be written as the magnetic field of a dipole, where m=Iπa^2 z-hat Now, naively, I would want to calculate the flux over the flat area spanned by the loop of radius b. However, the solution is much simpler if one uses a spherical cap of fixed radius R, which is bounded by the same loop of radius b. Hilarious. Because I never would have thought of that... So, the answer is the same no matter how you calculate, but this basically uses the idea that the flux through any surface, bounded by the same line, is the same. Clever. I mean, I knew that, but I don't see many examples like this... Can anyone provide another example of a clever use of math to help solve a difficult E&M problem?