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Question

Show that the radius of the orbit in hydrogen atoms varies as $$\displaystyle { n }^{ 2 }$$, where n is the principal quantum number of the atom.


Solution

The centripetal acceleration of the electron in a Bohr's atom is due to the Coulomb attraction of electron towards nucleus.
Thus, $$\dfrac{mv^2}{r}=\dfrac{1}{4\pi\epsilon_0}\dfrac{Ze^2}{r^2}$$
Also Bohr's postulate states, $$mvr=\dfrac{nh}{2\pi}$$
Substituting $$v$$ from second equation into the first gives, 
$$r\propto n^2$$

Physics

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