In the circuit shown below, inductances L₁, L₂ and L₃ are identical. The total magnetic energy stored in the circuit is U_t and the energy stored in the inductor L₂ is U_L2. Find the ratio U_t/U_L2.

The circuit consists of one inductor L₁ connected in series with two identical inductors L₂ and L₃ connected in parallel.

Let each inductance be equal to L.

The equivalent inductance of the parallel combination is:

L_p = (L × L)/(L + L)

L_p = L/2

Therefore total equivalent inductance:

L_eq = L + L/2

L_eq = 3L/2

Assume current I enters the circuit.

The current divides equally through L₂ and L₃ because both inductors are identical.

I₂ = I₃ = I/2

Total magnetic energy stored:

U_t = ½ L_eqI²

= ½ × (3L/2) × I²

= 3LI²/4

Energy stored in L₂:

U_L2 = ½ L(I/2)²

= LI²/8

Therefore:

U_t/U_L2

= (3LI²/4)/(LI²/8)

= 6

Thus, the total magnetic energy stored in the entire circuit is six times the energy stored in inductor L₂.

U_t/U_L2 = 6