You’ve probably heard the term magnetic gradient before, but have you ever seen one? Now you can in [supermagnetman’s] video, below. The key is to use very fine (2 micron) iron filings and special silicone oil. The video is a good mix of whiteboard lectures and practical hands-on experimenting. Just watching him spin the iron filings in the bottle was entertaining. There’s sources in the video description for the oil and the filings if you want to replicate the demonstrations for a classroom or just for your own enjoyment.
It’s one thing to know the math behind magnetic fields. It’s another to be able to use them in practical applications. But a good understanding of the physical manifestation of the magnetic field coupled can help clarify the math and vice versa. There’s a lot of common sense explanations too. For example, the way the filings accelerate as they get closer to the magnet explains why the patterns form the way they do. Iron filings are a traditional way to “see” magnetic fields. Ask anyone who ever had a Wooly Willy.