Q. Consider a pair of coupled coils as shown in Figure of the text, with currents, voltages, and polarity dots as indicated. Show that the mutual inductance is L₁₂ = L₂₁ = M by following these steps:

(a) Starting at time t₀ with i₁(t0) = i₂(t0) = 0, maintain i₂ = 0 and increase i₁ until, at time t₁, i₁(t₁) = I₁ and i₂(t₁) = 0. Determine the energy accumulated during this time. Now maintaining i₁ = I₁, increase i2 until at time t₂, i₂(t₂) = I₂. Find the corresponding energy accumulated and the total energy stored at time t₂.

(b) Repeat the process in the reverse order, allowing the currents to reach their ?nal values. Compare the expressions obtained for the total energy stored and obtain the desired result.