Ah – critical insight: If the core originally had , its reluctance is 497 kA-t/Wb. Then flux would be (250/497k \approx 0.503 \ \textmWb), not 1.2 mWb. So the “desired” 1.2 mWb must have come from a different core or higher current. The problem as written is inconsistent – an excellent teaching point: always check if numbers make physical sense .
Percent change from Problem 2: [ \frac0.232 - 0.2010.201 \times 100 \approx +15.4% ] Fringing reduces reluctance → increases flux. Ignoring fringing underestimates performance. Solution 4 – Series-Parallel Circuit Step 1 – Reluctances (all (\mu = 1000 \mu_0))
Flux density: [ B = \frac\PhiA = \frac1.005\times 10^-35\times 10^-4 = 2.01 \ \textT ] Good – below saturation for typical iron. Solution 2 – With Air Gap (a) Core reluctance same as above: (\mathcalR_c \approx 398 \ \textkA-turns/Wb) Gap reluctance: [ \mathcalR g = \fracl_g\mu_0 A = \frac0.001(4\pi\times 10^-7)(5\times 10^-4) \approx 1.592 \times 10^6 \ \textA-turns/Wb ] Total reluctance: [ \mathcalR total = 3.98\times 10^5 + 1.592\times 10^6 = 1.99 \times 10^6 \ \textA-turns/Wb ]
