Question

The magnetic moment due to the motion of the electron in the nth energy state of a hydrogen atom is proportional to:

• A. n
• B. n²
• C. n³
• D. n⁰

Answer 1

The Correct Option is A

The magnetic moment (μ) of an electron in the nth energy state of a hydrogen atom is given by: 

μ = (eħ/2m_e) × l

where l is the orbital angular momentum quantum number (l = n-1 for hydrogen)

However, for circular orbits (which maximize the magnetic moment), we can derive:

μ_n = n(eħ/2m_e) = nμ_B

where μ_B is the Bohr magneton (fundamental constant)

Key relationships:

• Orbital radius r ∝ n²
• Velocity v ∝ 1/n
• Current I = e/T ∝ v/r ∝ 1/n³
• Magnetic moment μ = IA ∝ (1/n³) × n⁴ = n

The magnetic moment is directly proportional to n (the principal quantum number).

Answer 2

The magnetic moment due to the motion of an electron in a hydrogen atom is primarily a consequence of its orbital angular momentum. This can be described by the Bohr model, where the electron moves in orbits around the nucleus.

In the Bohr model of the hydrogen atom, the angular momentum \(L\) of an electron in the nth orbit is quantized and given by: \(L = n\hbar\), where \(\hbar\) is the reduced Planck's constant.

The magnetic moment \(\mu\) associated with this angular momentum is related through the equation: \(\mu = \dfrac{e}{2m}L\), where \(e\) is the electron charge and \(m\) is the mass.

Substituting the expression for \(L\), we have: \(\mu = \dfrac{e}{2m}(n\hbar)\).

Thus, the magnetic moment \(\mu\) is proportional to \(n\), making the correct answer \(n\).