An electron revolving in nᵗʰ Bohr orbit has magnetic moment μₙ. If μₙ ∝ nˣ, the value of x is

Solution

Physics sir · 4 steps

Write the magnetic moment in terms of angular momentum, and Bohr's rule hands you the n-dependence.

1. Orbit as a loop. The revolving electron acts as a current loop. Its current is I = e/T = ev/(2π r) and its area is A = π r², so μ = IA = evr/2.

   μₙ = evr/2 = e/2m (mvr) = e/2m L

2. Use Bohr's rule. Bohr quantisation says L = mvr = nh/2π. So μₙ = e/2m × nh/2π = n eh/(4π m).
3. Cross-check. From the radius and speed: r ∝ n² and v ∝ 1/n, so vr ∝ n. Both routes give the same result.
4. So. μₙ ∝ n¹, so x = 1, which is option (d).

The idea to keep. For any orbiting charge, μ = q/2mL. In a Bohr orbit L ∝ n, so the magnetic moment grows in direct proportion to n.

Answer from the NTA final answer key. Solution by PrepWiser.