A magnetic dipole of moment M and moment of inertia I is placed in two mutually perpendicular uniform magnetic fields B₁ and B₂. It is allowed to execute small oscillations about its stable equilibrium direction.
Its time period is:

Question

A magnetic dipole of moment M and moment of inertia I is placed in two mutually perpendicular uniform magnetic fields B₁ and B₂. It is allowed to execute small oscillations about its stable equilibrium direction.

Its time period is:

G1 Physics · Accepted Answer

The two perpendicular magnetic fields combine vectorially. Therefore, B e f f B_eff B e ​ f f = √(B₁²+B₂²). The stable equilibrium direction is along the resultant field. For a small displacement φ from this equilibrium direction, τ ≈ -M B e f f B_eff B e ​ f f φ. Hence, I d²φ/dt² + M B e f f B_eff B e ​ f f φ = 0. So, ω = √(M B e f f / I ) . B_eff/I). B e ​ f f / I ) . Therefore, T = 2π√(I/M B e f f ) B_eff) B e ​ f f ) = 2π√[I/{M√(B₁²+B₂²)}].