Q107 / 150 · MHT CET 2026 (15 April Shift 2) · Physics
In the hydrogen atom, radii of the first four Bohr orbits are related as
- A$1 : 2 : 3 : 4$
- B$\dfrac{1}{2} : \dfrac{1}{3} : \dfrac{1}{4}$
- Correct answerC$1 : 4 : 9 : 16$
- D$\dfrac{1}{4} : \dfrac{1}{9} : \dfrac{1}{16}$
Solution
The radius of the $n$-th Bohr orbit in a hydrogen-like atom is given by $r_n = \dfrac{n^2 h^2 \epsilon_0}{\pi m e^2 Z}$.
For a hydrogen atom, $Z = 1$, which implies $r_n \propto n^2$.
For the first four orbits, $n = 1, 2, 3, 4$.
The ratio of their radii is $r_1 : r_2 : r_3 : r_4 = 1^2 : 2^2 : 3^2 : 4^2 = 1 : 4 : 9 : 16$.
Answer: $1 : 4 : 9 : 16$
For a hydrogen atom, $Z = 1$, which implies $r_n \propto n^2$.
For the first four orbits, $n = 1, 2, 3, 4$.
The ratio of their radii is $r_1 : r_2 : r_3 : r_4 = 1^2 : 2^2 : 3^2 : 4^2 = 1 : 4 : 9 : 16$.
Answer: $1 : 4 : 9 : 16$
Hence, the correct answer is option: 'C'.