Question

The figure shows standing de Broglie waves due to the revolution of electron in a certain orbit of hydrogen atom. Then the expression for the orbit radius is (all notations have their usual meanings)

• A. \(\frac{h^2\in_0}{\pi me^2}\)
• B. \(\frac{4h^2\in_0}{\pi me^2}\)
• C. \(\frac{9h^2\in_0}{\pi me^2}\)
• D. \(\frac{36h^2\in^{2}_0}{\pi me^2}\)

Answer 1

The Correct Option is D

Given the number of stationary waves is 6, i.e., the quantum number:

\[ n = 6 \]

The radius of the orbit is determined by the formula:

\[ r = \frac{n^2 \varepsilon_0^2 h^2}{\pi me^2} \]

Substituting \( n = 6 \) into the equation:

\[ r = \frac{6^2 \varepsilon_0^2 h^2}{\pi me^2} = \frac{36 \varepsilon_0^2 h^2}{\pi me^2} \]

Where:

• \( \varepsilon_0 \) is the permittivity of free space,
• \( h \) is Planck's constant,
• \( m \) is the mass of the electron,
• \( e \) is the charge of the electron.