Clase 13 de abril

$$
U_{rho}[\rho,\Phi_{ext}] =\sum_{i} q_i V(\vec{r}_i)
$$

$$
U_{total}[\rho] = \sum_{i <j} \frac{q_i q_j}{4\pi \varepsilon_0 |\vec{r}_i-\vec{r}_j|}
$$

$$
U_{total}[\rho] = \frac{1}{2}\sum_{i\neq j} \frac{q_i q_j}{4\pi \varepsilon_0 |\vec{r}_i-\vec{r}_j|}
$$

$$
U_{total}[\rho] = \frac{1}{2}\iint\frac{\rho(\vec{r})\rho(\vec{r}')dV dV'}{4\pi \varepsilon_0 |\vec{r}_i-\vec{r}_j|}=\frac{1}{2}\int \rho(\vec{r})\Phi(\vec{r})dV
$$

\begin{eqnarray}
U_{total}[\rho] &=& \frac{1}{2}\int -\varepsilon_0 (\nabla^2 \Phi(\vec{r}))\Phi(\vec{r})dV\\
&=&
\frac{1}{2}\int -\varepsilon_0 (\nabla\cdot\nabla \Phi(\vec{r}))\Phi(\vec{r})dV\\
&=&
\frac{1}{2}\int -\varepsilon_0 \nabla\cdot(\Phi(\vec{r})\nabla \Phi(\vec{r}))dV+
\frac{1}{2}\int \varepsilon_0 |\nabla \Phi(\vec{r})|^2 dV\\
&=&
\frac{1}{2}\int -\varepsilon_0 |\vec{E}|^2dV
\end{eqnarray}

$$
\vec{F}=\int \rho(\vec{r})\vec{E}(\vec{r}) dV=
\vec{F}=\int \rho(\vec{r}) dV \vec{E}(0) =
$$

$$
\vec{\cal T}=\int \vec{r}\times\rho(\vec{r})\vec{E}(\vec{r}) dV\approx
(\int \rho(\vec{r})\vec{r} dV) \times\vec{E}(0)=\vec{p}\times \vec{E}(0)
$$

$$
(\int \rho(r) r_i r_m dr) (\partial_m \partial_j \Phi) \epsilon_{ijk} \check{e}_k
$$

$$
\vec{E}(\vec{r})=\vec{E}(0) + \vec{r}\cdot \nabla \vec{E}
$$

$$
\vec{E}(\vec{r})= \frac{3 ((\vec{r}-\vec{r}')\cdot \vec{p})(\vec{r}-\vec{r}')/|\vec{r}-\vec{r}'|^2-\vec{p}}{4\pi\varepsilon_0|\vec{r}-\vec{r}'|^3}
$$

$$
\vec{E}(0)= \frac{3 (\check{r}'\cdot \vec{p})\check{r}'-\vec{p}}{4\pi\varepsilon_0|\vec{r}'|^3}
$$

$$
\vec{r}\cdot \nabla\vec{E}(0)= ...
$$