Question

A particle having mass $m$ and charge $(-q)$ moves along an ellipse around a fixed charge $Q$ such that its maximum and minimum distances from the fixed charge are $r_1$ and $r_2$ respectively. Show that the angular momentum $L$ of this particle about $Q$ is a $\times \cdot 10^{4}Ns$, then find $a$. $(Q=2C,q=1C,r_1=3m$, $r_2=1m,m=0.3kg$ )

Answer 1

Answer: 9.00

(1) Angular momentum conservation about charge $+Q$
$L=m{V}_1{r}_1=m{V}_2{r}_2...(1)$
(2) Energy conservation
$\frac{1}{2}m{V}_1{}^2-\frac{qQ}{4\pi {\epsilon }_0{r}_1}=\frac{1}{2}m{V}_2{}^2-\frac{qQ}{4\pi {\epsilon }_0{r}_2}...(2)$
from (1) and (2)
$V_1=\sqrt{\frac{r_{2}Qq}{2\pi {\epsilon }_{0}m\left(r_1+r_2\right)r_1}}$
(3) $L=m{V}_1{r}_1$
$L=\sqrt{\frac{m{r}_1{r}_{2}Qq}{2\pi {\epsilon }_0\left(r_1+r_2\right)}}$