To determine how the angular momentum of an electron in an orbit of a hydrogen atom relates to the radius \(R\), we start with the quantized angular momentum condition for hydrogen:
The angular momentum \(L\) of an electron in orbit is given by:
\(L = n \hbar\)
where \(n\) is the principal quantum number (an integer) and \(\hbar\) is the reduced Planck's constant.
For a hydrogen atom, the radius of the nth orbit is given by the Bohr radius formula:
\(R_n = n^2 \frac{\hbar^2}{m e^2}\)
where
\(m\) is the mass of the electron,
\(e\) is the charge of the electron, and
\(\hbar\) is the reduced Planck’s constant.
Solving the relationship between angular momentum and radius:
Since we have \(R \propto n^2\) and \(L = n \hbar\), given \(L \propto n\), if we solve for \(n\) in terms of \(R\), we have \(n \propto \sqrt{R}\).
Thus, the relation of angular momentum to radius \(R\) is:
