For hydrogen-like atoms and ions, the ratio of the magnetic dipole moment to the angular momentum ( l = m v r ) is a universal constant. Determine the value of this constant.

Step-by-step solution

The magnetic dipole moment of an electron revolving in a circular orbit is given by M = I A . The equivalent current is I = e T = e v 2 r and the area of the orbit is A = r^2 . M = ( e v 2 r ) ( r^2) = e v r 2 The angular momentum of the electron is L = m_e v r . The ratio of the magnetic dipole moment to the angular momentum is: M L = e v r 2 m_e v r = e 2 m_e Substituting the values e = 1.6 10⁻¹⁹ C and m_e = 9.1 10⁻³¹ kg : M L = 1.6 10⁻¹⁹ 2 9.1 10⁻³¹ = 1.6 18.2 10¹² 8.8 10¹⁰ C kg ⁻¹