Question

In an atom, two electrons move around the nucleus in circular orbits of radii $R$ and $4R$. The ratio of the time taken by them to complete one revolution is :
(neglect electric interaction)

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

Answer 1

The correct option is C $1:8$

If the electron is moving with a velocity $v$ around the nucleus the time taken by it to complete one revolution would be given by
$T=\frac{2\pi r}{v}$
Now, from Bohr's theory we have
$r\propto n^2$
$v\propto \frac{1}{n}$
$\Rightarrow T\propto n^3$

$T^2\propto n^6$ or $T^2\propto r^3$
${\left(\frac{T_1}{T_2}\right)}^2={\left(\frac{R}{4R}\right)}^3=\frac{1}{64}$
$\frac{T_1}{T_2}=\frac{1}{8}$