Supposedly I keep a rod of mass 3 kg and length 2 meters on a smooth desk, such that masses of 1 kg and 2 kg are stuck firmly to top and bottom of the rod respectively. Now, the center of mass of the resultant body is changed, from what I've calculated, it has shifted 1/6 meters below the center...

The differential equation for a optical fiber with a refractive index $n(r)$ is given as $$\nabla^{2}_{\perp}A(r,\theta)+(k^{2}n(r)^2-\beta^2)A(r,\theta)=0.$$ which is separable in cylindrical coordinates, $A(r,\theta)=R(r)\Theta(\theta)$. The radial DE follows as $$ r^{2}R''+rR'+[r^{2}(k^{2}n(...

First, to make it clear Stack Exchange does not allow animated avatars, the Can I have an animated profile picture? feature request has been declined by SE staff back in 2016. However, as of today, users who upload animated gif as avatar have the animation showing in their profile, e.g. this use...

In Peskin and Shroeder, for a local $U(1)$ transformation, the comparator operator is expanded as: \begin{equation} U(x+\epsilon n, x) = 1 -ie\epsilon n^{\mu}A_{\mu} + \mathcal{O}(\epsilon^2) \tag{15.5} \end{equation} for $\epsilon\rightarrow0$. I am not sure how one arrives at this expression, a...