24. Work Done Separating Coaxial Rings
Step 1: Conservation of Flux
The rings are perfectly conducting ($R=0$), so magnetic flux is conserved. Initially, they are infinitesimally close ($M \approx L$).
$$ \Phi_{\text{initial}} = L I_0 + M I_0 \approx 2 L I_0 $$
Finally, they are separated ($M=0$). Let final current be $I_f$.
$$ \Phi_{\text{final}} = L I_f $$
$$ 2 L I_0 = L I_f \implies I_f = 2 I_0 $$
Step 2: Energy Calculation
Work done is the change in potential energy ($W = U_f – U_i$).
$$ U_i = \frac{1}{2} (2LI_0) I_0 + \frac{1}{2} (2LI_0) I_0 = 2 L I_0^2 $$
$$ U_f = 2 \times \left( \frac{1}{2} L I_f^2 \right) = L (2I_0)^2 = 4 L I_0^2 $$
Work Done = $4 L I_0^2 – 2 L I_0^2 = 2 L I_0^2$