Answer:

Introduction:
The magnetic dipole moment of an electron is the product of its charge and the distance between the two opposite poles of the electron. The minimum magnetic dipole moment of an electron in a hydrogen atom can be determined using the Bohr model of the atom.

Bohr Model:
The Bohr model of the atom describes the hydrogen atom as a positively charged nucleus surrounded by a single electron in a circular orbit. The electron is held in its orbit by the electrostatic attraction between the positive nucleus and the negative electron.

Quantization of Angular Momentum:
In the Bohr model, the electron in the hydrogen atom can only occupy certain discrete energy levels. This is due to the quantization of angular momentum, which states that the angular momentum of the electron must be an integer multiple of the reduced Planck constant, ħ.

Quantization of Magnetic Moment:
The magnetic dipole moment of the electron is proportional to its angular momentum. Therefore, the quantization of angular momentum also leads to the quantization of the magnetic dipole moment of the electron. The magnitude of the magnetic dipole moment of the electron is given by:

μ = (e/2m) * L

where e is the charge of the electron, m is its mass, and L is its angular momentum.

Minimum Magnetic Dipole Moment:
The minimum value of the magnetic dipole moment of the electron in the hydrogen atom occurs when the electron is in its ground state. In this state, the electron's angular momentum is given by:

L = ħ

Substituting this value into the equation for the magnetic dipole moment yields:

μ = (e/2m) * ħ

Simplifying this expression gives:

μ = eh/2πm

Therefore, the minimum magnetic dipole moment of the electron in the hydrogen atom is eh/2πm.

Conclusion:
The minimum magnetic dipole moment of the electron in the hydrogen atom is eh/2πm. This value is derived from the quantization of angular momentum and the quantization of the magnetic dipole moment of the electron.

Eh/4Pi m