Question

A particle of charge $-q$ and mass $m$ moves in a circular orbit of radius $r$ about a fixed charge $+Q$. The relation between the radius of the orbit $r$ and the time period $T$ is:

• A. $r=\frac{Qq}{16{\pi }^2{ϵ}_{0}m}{T}^3$
• B. $r^3=\frac{Qq}{16{\pi }^3{ϵ}_{0}m}{T}^2$
• C. $r^2=\frac{Qq}{16{\pi }^3{ϵ}_{0}m}{T}^3$
• D. $r^2=\frac{Qq}{16\pi {ϵ}_{0}m}{T}^3$

Answer 1

The correct option is B $r^3=\frac{Qq}{16{\pi }^3{ϵ}_{0}m}{T}^2$

The centrifugal force acting on charge $q$ is:
$F=\frac{kqQ}{r^2}$
$\Rightarrow \frac{m{v}^2}{r}=\frac{kqQ}{r^2}$
$\Rightarrow V=\sqrt{\frac{kqQ}{rm}}$
Now, $V=\frac{2\pi r}{T}$
$\Rightarrow \frac{2\pi r}{T}=\sqrt{\frac{qQ}{4\pi {\epsilon }_{0}rm}}$
$\Rightarrow \frac{4{\pi }^2{r}^2}{T^2}=\frac{qQ}{4\pi {\epsilon }_{0}rm}$
$\Rightarrow r^3=\frac{qQ{T}^2}{16{\pi }^3{\epsilon }_{0}m}$