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 of an electron in an atom is related to The magnetic moment of an electron in an atom is related to its angular momentum. 

For a hydrogen atom, in the n^th energy state, the angular momentum is proportional to the principal quantum number n. As a result, the magnetic moment is also proportional to n.

Thus, the correct answer is (1), as the magnetic moment is directly proportional to n.its angular momentum. 

For a hydrogen atom, in the nth energy state, the angular momentum is proportional to the principal quantum number n. 

As a result, the magnetic moment is also proportional to n.