Physvillain

Considering a simple uniformly surface-charged massless sphere with radius $a$ and total charge $q$ (see Figure 1), energy and momentum of an electromagnetic field can be acquired: \begin{align} E_\text{EM} &=\frac{1}{2}\int \mathbf{E} \cdot \mathbf{D} \; d^3x = \frac{1}{2}\frac{q^2}{4\pi\epsilon_0}\frac{1}{a}, \\ \textbf{p}_\text{EM} &=\int \mathbf{E}\times \mathbf{H} \; d^3x = \frac{2}{3} \fra..

In classical physics, all physical quantities were considered to satisfy the commutation relation. For example, in classical mechanics, two quantities $A(x,p)$ and $B(x,p)$ have the commutative property of multiplication. But in the quantum mechanics, the commutation of $A$ and $B$ is not zero in general. \begin{equation}\tag{1} \left[ A(x, p), B(x, p) \right] \neq 0. \end{equation} The history ..

The operator form of the relativistic dispersion relation leads to the dynamics of field, yielding the form of the Lagrangian \begin{equation} \mathcal{L} = -\frac{1}{4}F_{\mu\nu} F^{\mu\nu} + \frac{m^2}{2}A_\mu A^\mu \end{equation} for a massive (abelian) gauge boson $A_\mu$, and \begin{equation}\label{lagrangian_fermion}\tag{0} \mathcal{L} = i\bar{\psi} \gamma^\mu \partial_\mu \psi - m \bar{\p..

Case I. Conductor loop entering a magnetic field We've learned that induced electromotive force (EMF) is induced by changes in magnetic flux inside a closed loop, but this is actually nothing new that is naturally derived by known physics. Either magnetic or electric force can explain these induced EMF. We will explore these facts, as well as a new insight from relativity. First, consider the si..

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