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 expressed as: μ = (eħ/2m_e) × l, where l represents the orbital angular momentum quantum number (for hydrogen, l = n-1).

For circular orbits, which yield the maximum magnetic moment, the formula simplifies to: μ_n = n(eħ/2m_e) = nμ_B, where μ_B denotes the Bohr magneton, a fundamental constant.

Key proportionalities observed are:

• Orbital radius (r) is proportional to n².
• Velocity (v) is proportional to 1/n.
• Current (I), calculated as e/T, is proportional to v/r, thus proportional to 1/n³.
• Magnetic moment (μ), given by IA, is proportional to (1/n³) × n⁴, simplifying to n.

Therefore, the magnetic moment exhibits a direct proportionality to n, the principal quantum number.