Modern Physics 1 Question 40
37. The electron in a hydrogen atom makes a transition $n _1 \rightarrow n _2$, where $n _1$ and $n _2$ are the principal quantum numbers of two states. Assume the Bohr’s model to be valid. The time period of the electron in the initial state is eight times that in the final state. The possible values of $n _1$ and $n _2$ are
$(1998,2 M)$
(a) $n _1=4, n _2=2$
(b) $n _1=8, n _2=2$
(c) $n _1=8, n _2=1$
(d) $n _1=6, n _2=3$
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Solution:
- Time period, $T _n=\frac{2 \pi r _n}{v _n} \quad$ (in $n$th state)
$$ \begin{array}{lll} \text { i.e. } & T _n \propto \frac{r _n}{v _n} \\ \text { But } & r _n \propto n^{2} \\ \text { and } & v _n \propto \frac{1}{n} \\ & \text { Therefore, } & T _n \propto n^{3} \\ \text { Given } & T _{n _1} & =8 T _{n _2} \\ \text { Hence, } & n _1 & =2 n _2 \end{array} $$
