Determine the magnetic dipole moment associated with the motion of...

The magnetic dipole moment of an electron in the $n$-th orbit of a hydrogen atom is given by $M = \dfrac{e v r}{2}$.

From Bohr's quantization condition, the angular momentum is $L = m_e v r = \dfrac{n h}{2 \pi}$.

Thus, $v r = \dfrac{n h}{2 \pi m_e}$.

Substituting this into the expression for magnetic moment:

$M = \dfrac{e}{2} \left( \dfrac{n h}{2 \pi m_e} \right) = \dfrac{n e h}{4 \pi m_e}$

For the ground state, $n = 1$.

$M = \dfrac{e h}{4 \pi m_e}$

Substituting the standard values $e = 1.6 \times 10^{-19} \text{ C}$, $h = 6.63 \times 10^{-34} \text{ J s}$, and $m_e = 9.1 \times 10^{-31} \text{ kg}$:

$M = \dfrac{1.6 \times 10^{-19} \times 6.63 \times 10^{-34}}{4 \times 3.14 \times 9.1 \times 10^{-31}}$

$M \approx 9.27 \times 10^{-24} \text{ A m}^2$