Question

In two $H$ atoms $X$ and $Y$ the electrons move around the nucleus in circular orbits of radius $r$ and $4r$ respectively. The ratio of the times taken by them to complete one revolution is :

• A. $1:4$
• B. $1:2$
• C. $1:8$
• D. $2:1$

Answer 1

The correct option is C $1:8$

Time of resolution is calculated as

given, $r_1=r;r_2=4r$

$T=\frac{2\pi R}{V};$ $v=$ velocity

We know the radius but velocity is unknown

for Orbit 1: $\frac{m{V}^2}{r}=\frac{kZ{e}^2}{r^2}$

$v=\frac{c}{\sqrt{r}}$

for Orbit 2: $\frac{m{V}^2}{4r}=\frac{kZ{e}^2}{16r^2}$

$v=\frac{c}{2\sqrt{r}}$

$T_1=\frac{2\pi r}{c/\sqrt{r}}$

$T_2=\frac{8\pi r}{c/\sqrt{r}}$

$T_1:T_2=1:8$