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Question

(C) The time period $T$ for an electron in a circular orbit is given by $T = \frac{2 \pi r}{v}$.According to Bohr's theory,the radius $r \propto n^{2}

Vedclass · Accepted Answer

$1 : 8$

Vedclass · Answer

(C) The time period $T$ for an electron in a circular orbit is given by $T = \frac{2 \pi r}{v}$.According to Bohr's theory,the radius $r \propto n^{2}$ and the velocity $v \propto \frac{1}{n}$.Substituting these into the expression for $T$,we get $T \propto \frac{r}{v} \propto \frac{n^{2}}{1/n} = n^{3}$.Since $r \propto n^{2}$,we have $n \propto r^{1/2}$,so $T \propto (r^{1/2})^{3} = r^{3/2}$.Therefore,the ratio of the time periods is $\frac{T_{1}}{T_{2}} = \left(\frac{R_{1}}{R_{2}}\right)^{3/2}$.Given $R_{1} = R$ and $R_{2} = 4R$,we have $\frac{T_{1}}{T_{2}} = \left(\frac{R}{4R}\right)^{3/2} = \left(\frac{1}{4}\right)^{3/2} = \frac{1}{8}$.The ratio is $1 : 8$.

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:

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