Question

Small identical balls with equal charges are fixed at vertices of regular polygon with side $a$. At a certain instant, one of the balls is released & a sufficiently long time interval later, the ball adjacent to the first released ball is freed. The kinetic energies of the released balls are found to differ by $K$ at a sufficiently long distance from the polygon. Determine the charge $q$ of each ball. If it is $\frac{1}{a}\times 10^{-4}C$. Find $a.(k=10$ Joule, side length $=1m)$

Answer 1

Answer: 6.00

$K{E}_1:$ Kinetic energy of $1^{\text{st }}$
$K{E}_2:$ Kinetic energy of $2^{\text{nd }}$
$K.E_1$
$=\frac{q^2}{4\pi {\epsilon }_0{r}_{12}}+\frac{q^2}{4\pi {\epsilon }_0{r}_{13}}+\dots +\frac{q^2}{4\pi {\epsilon }_0{r}_{1n}}$
$K.E_2$
$=\frac{q^2}{4\pi {\epsilon }_0{r}_{22}}+\frac{q^2}{4\pi {\epsilon }_0{r}_{23}}+\dots .+\frac{q^2}{4\pi {\epsilon }_0{r}_{2n}}$
$=\frac{q^2}{4\pi {\epsilon }_0{r}_{1n}}+\frac{q^2}{4\pi {\epsilon }_0{r}_{1,n-1}}+\dots +\frac{q^2}{4\pi {\epsilon }_0{r}_{13}}$
$K=K{E}_1-K{E}_2=\frac{q^2}{4\pi {\epsilon }_0{r}_{1\alpha }}=\frac{q^2}{4\pi {\epsilon }_{0}a}$
$q=\sqrt{4\pi {\epsilon }_{0}ak}$