Question

In an atom, two electrons complete three revolutions around the nucleus in circular orbit in time \(81t\) and \(192t\) respectively. The ratio of their radii will be (\(t\) is in second)

• A. \(9:16\)
• B. \(4:3\)
• C. \(27:64\)
• D. \(3:4\)

Hint:

In atomic orbits, always remember \( T \propto r^{3/2} \).

Answer 1

The Correct Option is A

Step 1: Time period for one revolution.
If an electron completes 3 revolutions in time \(T\), then time period is
\[ T_1 = \frac{81t}{3} = 27t, \quad T_2 = \frac{192t}{3} = 64t \]
Step 2: Relation between time period and radius.
For an electron in circular orbit around nucleus:
\[ T \propto r^{3/2} \]
Step 3: Taking ratio of radii.
\[ \left(\frac{r_1}{r_2}\right)^{3/2} = \frac{T_1}{T_2} \] \[ \left(\frac{r_1}{r_2}\right)^{3/2} = \frac{27}{64} \]
Step 4: Solving for radius ratio.
\[ \frac{r_1}{r_2} = \left(\frac{27}{64}\right)^{2/3} \] \[ \frac{r_1}{r_2} = \frac{9}{16} \]
Step 5: Conclusion.
The ratio of their radii is \(9:16\).