on 17-Nov-2023 (Fri)

Divergence of gradient: \(\begin{align}\nabla\cdot(\nabla T)=\frac{\partial^{2} T}{\partial x^{2}} +\frac{\partial^{2} T}{\partial y^{2}} +\frac{\partial^{2} T}{\partial z^{2}}\end{align}\)

Curl of gradient: The curl of a gradient is always zero:\(\nabla\times(\nabla T)=\mathbf{0}\)

Curl of curl: Curl-of-curl gives nothing new: \(\nabla\times(\nabla\times T)=\nabla(\nabla\cdot\mathbf{v})-\nabla^{2}\mathbf{v}\)

Integrational Form of Maxwell's Equations in Vacuum:

\(\begin{array}{rrl}\\(\text{i})&\qquad\displaystyle\oint_{S}\mathbf{E}\cdot \mathrm{d}\mathbf{a}&=&0\qquad &\text{(Gauss' law)}\\\\(\text{ii})&\qquad\displaystyle\oint_{S}\mathbf{B}\cdot \mathrm{d}\mathbf{a}&=&0\qquad &\text{(Gauss's law for magnetism)}\\\\(\text{iii})&\qquad\displaystyle\oint_{L}\ \mathbf{E} \cdot \mathrm{d}\boldsymbol{\mathbf{r}}&=&\displaystyle - \frac {\partial}{\partial t}\int_{S}\mathbf{B}\cdot\mathrm{d}\mathbf{a} \qquad &\text{(Faraday's law)}\\\\(\text{iv})&\qquad\displaystyle\oint_{L}\ \mathbf{B} \cdot \mathrm{d}\boldsymbol{\mathbf{r}}&=&\displaystyle\mu_0 \varepsilon_0 \frac{\partial}{\partial t}\int_{S}\mathbf{E}\cdot\mathrm{d}\mathbf{a}\qquad &\text{(Ampere's law with Maxwell's correction)}\\\end{array}\)

The Plane wave form of Electromagnetic wave is

\(\begin{array}{rl}\vec{E}&=&E_{0}\exp[i(kz-\omega t)]\cdot\hat{x}\\\vec{B}&=&B_{0}\exp[i(kz-\omega t)]\cdot\hat{y}\end{array}\)

where
- \(E_{0}\) is the amplitude of the electric field.
- \(B_{0}\) is the amplitude of the magnetic field.

- We also have \(B_{0}=\frac{k}{\omega}E_{0}=\frac{1}{c}E_{0}=\sqrt{\mu_{0}\varepsilon_{0}}E_{0}\)
- \(k\) is the wave number, or wave vector, representing the propagation direction.
- \(\omega\) is the angular frequency.
- \(\hat{x},\hat{y}\) is the polarization vector, representing the polarization direction.