User:Vvv444/sandbox2

$$\begin{align} & 4\pi {{r}^{2}}E\left( rR \right)=4\pi \left( \frac{4}{3}\pi {{R}^{3}}\rho -Q \right)\Rightarrow E\left( r>R \right)=\left( \frac{4}{3}\pi {{R}^{3}}\rho -Q \right)\frac{1} \\ & \\  & \varphi \left( r>R \right)=\int_{r}^{\infty }{E\left( {r}'>R \right)d{r}'}=\int_{r}^{\infty }{\left( \frac{4}{3}\pi {{R}^{3}}\rho -Q \right)\frac{1}d{r}'}=\left( \frac{4}{3}\pi {{R}^{3}}\rho -Q \right)\frac{1}{r} \\ & \varphi \left( r<R \right)=\varphi \left( R \right)+\int_{r}^{R}{E\left( {r}'<R \right)d{r}'}=\left( \frac{4}{3}\pi {{R}^{3}}\rho -Q \right)\frac{1}{R}+\int_{r}^{R}{\left( \frac{4}{3}\pi r\rho -\frac{Q} \right)d{r}'}= \\ & =\left( \frac{4}{3}\pi {{R}^{3}}\rho -Q \right)\frac{1}{R}+\frac{2}{3}\pi \rho \left( {{R}^{2}}-{{r}^{2}} \right)+Q\left( \frac{1}{R}-\frac{1}{r} \right)= \\ & =2\pi {{R}^{2}}\rho -\frac{2}{3}\pi \rho {{r}^{2}}-\frac{Q}{r} \\ \end{align}$$

$$\begin{align} & \varphi \left( a \right)=\varphi \left( b \right)+\int_{a}^{b}{E\cdot dr} \\ & in\,\,many\,\,cases\,\,we\,\,choose\,\,\varphi \left( \infty \right)=0 \\ & \\  & \varphi \left( rR \right)\cdot d{r}'}+\int_{r}^{R}{E\left( {r}'<R \right)\cdot d{r}'}= \\ & =\varphi \left( R \right)+\int_{r}^{R}{E\left( {r}'<R \right)\cdot d{r}'} \\ \end{align}$$