Monthly Archives: May 2024

Atomic Structure of Materials Materials For Devices Diffraction Microstructure Mechanical Behavior of Materials Closer examination showed that each stripe was like a little stair on a staircase (will turn out to be of magnitude \(|\textbf b|\)); lattice planes of that … Continue reading →

This can be used as a reference to estimate the energies of other particles in infinite potential wells of other widths via \(E_n=\frac{\hbar^2}{2m}\left(\frac{n\pi}{L}\right)^2\). Also, although ground (GND) in electrical engineering is where \(\phi=V=0\), here “ground state” is actually above electrical … Continue reading →

Problem #\(1\): Let \(\phi_0(\textbf x)\) be the fundamental solution of the Helmholtz operator on \(\textbf x\in\textbf R^3\), that is: \[\left(-\biggr|\frac{\partial}{\partial\textbf x}\biggr|^2+k_{\phi}^2\right)\phi_0(\textbf x)=\delta^3(\textbf x)\] subject to the asymptotic homogeneous boundary conditions \(\lim_{|\textbf x|\to\infty}\phi_0(\textbf x)=\lim_{|\textbf x|\to\infty}\frac{\partial\phi_0}{\partial r}=0\). Compute \(\phi_0\). Solution #\(1\): Taking … Continue reading →

The purpose of this post is to compile solutions to select exercises from Steven Strogatz’s textbook Chaos and Nonlinear Dynamics. Chapter #\(1\): Chapter #\(2\): Chapter #\(3\): Chapter #\(4\): Chapter #\(5\): Chapter #\(6\): Chapter #\(7\): Chapter #\(8\):

The purpose of this post is to review how the fields of an electrostatic dipole \(\boldsymbol{\pi}\) and magnetostatic dipole \(\boldsymbol{\mu}\) arise. For the electrostatic dipole, “fields” means the electrostatic potential \(\phi\) and by extension the electrostatic field \(\textbf E=-\partial\phi/\partial\textbf x\) … Continue reading →

Given a flow field \(\textbf v(\textbf x,t)\), the vorticity \(\boldsymbol{\omega}\) of \(\textbf v\) is defined by taking its curl \(\boldsymbol{\omega}:=\frac{\partial}{\partial\textbf x}\times\textbf v\). For a flow field rotating rigidly with angular velocity vector \(\boldsymbol{\omega}_0\) so that \(\textbf v=\boldsymbol{\omega}_0\times\textbf x\). The vorticity … Continue reading →

Often in biochemistry, if a single substrate \(\text S\) needs to be become a product \(\text P\) via a chemical reaction of the form \(\text S\to \text P\). Assuming this is a first-order elementary chemical reaction, it would merely have … Continue reading →

The free, dispersionless wave equation on \((ct,\textbf x)\in\textbf R\times \textbf R^d\) is: \[\frac{\partial^2\psi}{\partial (ct)^2}-\frac{\partial^2\psi}{\partial|\textbf x|^2}=0\] or more simply as \(☐^2\psi=0\), where the d’Alembert operator is defined by \(☐^2:=\frac{\partial^2}{\partial (ct)^2}-\frac{\partial^2}{\partial|\textbf x|^2}=\partial^{\mu}\partial_{\nu}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\) with the usual metric \(\eta=\text{diag}(1,-1,-1,-1)\) defining the hyperbolic geometry of … Continue reading →

No one is perfect, and materials are the same. Technical ceramics like alumina, etc. are generally may be … but are often too brittle (i.e. have too low fracture toughness\(K^*\)). (maybe make a 3D Ashby materials selection scatter plot). The … Continue reading →