A magnetic dipole of moment M is slowly rotated from its stable equilibrium orientation to its unstable equilibrium orientation in a uniform magnetic field B.
The minimum external work required is:

Question

A magnetic dipole of moment M is slowly rotated from its stable equilibrium orientation to its unstable equilibrium orientation in a uniform magnetic field B. The minimum external work required is:

G1 Physics · Accepted Answer

At stable equilibrium, θ_i = 0, so U i U_i U i ​ = -MB. At unstable equilibrium, θ_f = π, so U f U_f U f ​ = +MB. For slow rotation, there is no change in kinetic energy. Hence external work is W e x t W_ext W e ​ x t = ΔU = U f U_f U f ​ - U i U_i U i ​ = MB - (-MB) = 2MB.