Question

Obtain an expression for the Bohr magneton.

Answer 1

Let an electron move in a circular orbit of radius r = ℏ²/me² around a proton. 

Assume that the z-component of the angular momentum is L_z = ℏ. Equating Lz to the classical angular momentum 

L_z = ℏ  = m_er²ω     ....(1) 

An electron orbiting the proton with frequency v = ω/2π constitutes a current 

i = ωe/2π

The magnetic moment μ₀ produced is equal to this current multiplied by the area enclosed. 

μ_B = i A = ωe. (πr²/2π)  ......(2) 

Using (1) in (2) 

μ_B = eℏ/2m_e   ....(3) 

μ_B is known as Bohr magneton. 

Electron with total angular momentum [j(j + 1)]^1/2ℏ has a magnetic moment μ = [j(j + 1)]^1/2μ_B 

The z-component of the magnetic moment is μ_J = m_Jμ_B where m_Jℏ is the z-component of the angular momentum.