Concepts Of Physics MCQ Edition [Volume 2]PhysicsBohr's Model and Physics of the Atom
Imagine a neutron and an electron bound to each other by gravitational force. Assuming Bohr's quantization rule for angular momentum is applicable in this scenario, determine the expression for the energy of the neutron-electron system.
Options
- A- 2 ^2 G^2 m_n m_e^2 n^2 h^2
- B- 4 ^2 G^2 m_n^2 m_e^3 n^2 h^2
- C- ^2 G^2 m_n^2 m_e^3 2 n^2 h^2
- D- 2 ^2 G^2 m_n^2 m_e^3 n^2 h^2
Correct answer
D. - 2 ^2 G^2 m_n^2 m_e^3 n^2 h^2
Step-by-step solution
The gravitational force provides the necessary centripetal force for the electron to revolve around the neutron. G m_n m_e r^2 = m_e v^2 r v^2 = G m_n r According to Bohr's quantization rule for angular momentum: m_e v r = n h 2 r = n h 2 m_e v Substituting r in the expression for v^2 : v^2 = G m_n 2 m_e v n h v = 2 G m_n m_e n h The total energy of the system is the sum of kinetic and potential energy, which is equal to the negative of the kinetic energy: E = -K = - 1 2 m_e v^2 E = - 1 2 m_e ( 2 G m_n m_e n h )^2