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(D) The radius of the $n^{th}$ orbit is given by $r_n \propto n^2$.The velocity of the electron in the $n^{th}$ orbit is given by $v_n \propto \frac{1

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$16 : 81$

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(D) The radius of the $n^{th}$ orbit is given by $r_n \propto n^2$.The velocity of the electron in the $n^{th}$ orbit is given by $v_n \propto \frac{1}{n}$.The centripetal acceleration $a_n$ is given by $a_n = \frac{v_n^2}{r_n}$.Substituting the proportionalities: $a_n \propto \frac{(1/n)^2}{n^2} = \frac{1}{n^4}$.For the $M$ shell, $n = 3$. Thus, $a_M \propto \frac{1}{3^4} = \frac{1}{81}$.For the $L$ shell, $n = 2$. Thus, $a_L \propto \frac{1}{2^4} = \frac{1}{16}$.The ratio of the initial ($M$ shell) to the final ($L$ shell) centripetal acceleration is $\frac{a_M}{a_L} = \frac{1/81}{1/16} = \frac{16}{81}$.

The electron in a hydrogen atom makes a transition from the $M$ shell to the $L$ shell. The ratio of the magnitudes of the initial to the final centripetal acceleration of the electron is:

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