Question

If $E$ and ${J_n}$ are the magnitudes of total energy and angular momentum of an electron in the nth Bohr orbit respectively, then,
(A) $E \propto J_n^2$
(B) $E \propto \dfrac{1}{{J_n^2}}$
(C) $E \propto {J_n}$
(D) $E \propto \dfrac{1}{{{J_n}}}$

Answer 1

Hint: In this question, it is given that the magnitude of the total energy and the angular momentum of the electron is in the nth Bohr orbit, so the relation between the total energy and the angular momentum is given by the Bohr’s theory.

Complete step by step answer:
Given that,
The total energy of the electron in the nth Bohr orbit is $E$.
The angular momentum of an electron in the nth Bohr orbit is ${J_n}$.
According to the Bohr theory,
When the electron is in the nth Bohr orbit, then the total kinetic energy is inversely proportional to the square of the distance between them. That is given by,
$E \propto \dfrac{1}{{{n^2}}}\,.................\left( 1 \right)$
And then, the linear momentum of the electron is inversely proportional to the distance between them. And the angular momentum of the electron is directly proportional to the distance between them.
Here, the angular momentum is required, so the angular momentum of the electron is directly proportional to the distance between them is given by,
${J_n} \propto n\,.................\left( 2 \right)$
By substituting the equation (2) in the equation (1), then the equation (1) is written as,
$E \propto \dfrac{1}{{J_n^2}}$
Thus, the above equation shows the relation between the energy of the electron and the angular momentum of the electron in the nth Bohr orbit.

Hence option (B) is the correct answer.

Note:
This solution can also be solved by using another method. By equation the Bohr model equation with the Coulomb force equation. By this we can get the kinetic energy and then by adding the kinetic energy and the potential energy, the relation between the energy of the electron, and the angular momentum of the electron can be determined.