Physics

A classic high-school physics problem: a charged particle enters a bounded uniform magnetic field perpendicularly, and we ask how strong the field must be so the particle “just exits through the right boundary.” The standard solution ignores gravity, and the trajectory is a clean circular arc. But what happens if gravity is included and the problem is flipped — the field is given, and the width is what we seek? The answer is: the circle becomes a cycloid, and the problem can still be solved elegantly — in two quite different ways. This post documents that extended problem (B6 Extension) and its two solutions. ...

High-school physics tells us that the displacement in simple harmonic motion (SHM) is a sinusoidal function. But textbooks never explain why — they assume a solution of the form $x(t)=\sin(\cdots)$, substitute it into the equation, verify that it works, and call it a day. That is a working backwards from the unknown to the known approach, and it has always left me uneasy. So I decided to do the opposite: starting from Hooke’s law as the sole axiom, I would analyse the evolution of velocity step by step and see whether I could derive the unknown from the known — letting that sine (or cosine) emerge on its own. This post records my two attempts: the first using brute-force infinite subdivision, the second using matrices and geometry. From complicated to clean, I ended up colliding head-on with Euler’s formula. ...

I have been systematically studying electromagnetism, tracing the full arc from electric fields, magnetic fields, and circuits through field energy, LC oscillations, and power generation. This post organises those scattered notes into a coherent framework — partly as a personal review, partly in the hope that it helps others working through the same material. The derivations involve quite a few equations, but the core is really just one thread: electricity and magnetism are two sides of the same coin, and energy flows through the field. ...