Physics - Modern Physics Question with Solution | TestHub

The formula to calculate the radius $\left(r\right)$ of any $n^{th}$ Bohr's orbit is given by

$r=r_0\frac{n^2}{Z}...\left(1\right)$

where, $r_0$ is Bohr's radius, $n$ is the principal quantum number and $Z$ is the atomic number.

Thus, for Helium ion,

$$\begin{array}{rl}{r}_1& =r_0\frac{2^2}{2}\\ & =2r_0...\left(2\right)\end{array}$$

And, for the Beryllium ion,

$$\begin{array}{rl}{r}_2& =r_0\frac{4^2}{4}\\ & =4r_0...\left(3\right)\end{array}$$

Divide equation (3) by equation (2) to obtain the required ratio.

$$\begin{array}{rl}\frac{r_2}{r_1}& =\frac{4r_0}{2r_0}\\ & =2\end{array}$$

Hence, $x=2$.