In the given circuit below inductance values of $$L_1$$, $$L_2$$ and $$L_3$$ are same. The magnetic energy stored in the entire circuit is $$(U_t)$$ and that stored in the $$L_2$$ inductor is $$(U_l)$$. $$U_t / U_l$$ is _______. (Ignore the mutual inductance if any)

Let each inductance = L and current entering the circuit = I

step 1: equivalent of parallel part

L₂ and L₃ are in parallel:

$$L_{eq}=\frac{L\cdot L}{L+L}=\frac{L}{2}$$

step 2: total inductance

L₁ is in series with this:

$$L_{total}=L+\frac{L}{2}=\frac{3L}{2}$$

step 3: total energy

$$U_t=\frac{1}{2}L_{total}I^2=\frac{1}{2}\cdot\frac{3L}{2}I^2=\frac{3}{4}LI^2$$

step 4: current division

current splits equally in L₂ and L₃:

$$I_2=I_3=\frac{I}{2}$$

step 5: energy in L₂

$$U_1=\frac{1}{2}L\left(\frac{I}{2}\right)^2=\frac{1}{2}L\cdot\frac{I^2}{4}=\frac{1}{8}LI^2$$

step 6: ratio

$$\frac{U_t}{U_1}=\frac{\frac{3}{4}LI^2}{\frac{1}{8}LI^2}=6$$