2018-10-09

Curl of a vector field

$\vec{\nabla} \times \vec{E} = \left | \begin{array}{cccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial }{\partial z} \\ {E_x} & {E_y} & {E_z} \end{array} \right |$

$\sum (\vec{\nabla}\times\vec{A})\cdot{\rm d \vec{S}}$
evaluate the curl of a vector field at a point and dot it with some unit vector, the result is the line integral per unit area of the vector field around an infinitesimal closed loop that is perpendicular to the unit vector
Electric monopoles
$V= \frac{q}{4\pi\epsilon_0r}$
$\vec{E} = \frac{q}{4\pi\epsilon_0r^2}\hat{r}$
with radial field lines and spherical equipotential surfaces
Electric dipoles
$V(r,\theta) = \frac{\vec{p}\cdot\hat{r}}{4\pi\epsilon_0r^2}$
$\vec{E}(\vec{r}) = -\vec{\nabla}V = -[\hat{r}\frac{\partial V}{\partial r} + \hat{\theta}\frac{1}{r}\frac{\partial V}{\partial \theta}]$

which evaluates to

$\vec{E}(r, \theta) = \frac{p}{4\pi\epsilon_0r^3}(\hat{r}2\cos\theta + \hat{theta}\sin \theta)$

if $p = p_0\sin\omega t$ then the $\vec{E}$ field oscillates and generate electric magnetic waves

torque on a diploe in a uniform field
$\vec{G} = \vec{p}\times\vec{E}$
energy of a dipole in uniform field
$U = -\vec{p}\cdot\vec{E}$
force on a dipole on a general electric field: taylor expand around the $E_-$ for $E_+$

$\vec{F} = (\vec{p}\cdot\vec{\nabla})\vec{E}$

using $\vec{\nabla}\times\vec{E} = 0$ and that $\vec{p}$ is time independent
$\vec{F} = \vec{\nabla}(\vec{p}\cdot\vec{E})$
which interestingly is
$\vec{F} = - \vec{\nabla}(U)$

2018-10-09

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