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The question is as follows:

For the core shown in Fig. 3.20, it is required to produce a flux of 2 mWb in the limb CD. The entire core has a rectangular cross section of 2cm × 2cm. The magnetizing coil has 800 turns. The relative permeability of the material is 1200. Calculate the amount of magnetizing current required.

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My question is that at point B and E the flux is getting distributed in two branches. Let's consider point B. At point B the flux gets divided into phi 1 and phi 2 . As per the solution, the length of for the reluctance of total flux (phi) is taken as 26 cm . Why the length is taken from the middle point of branch BE. The flux will start dividing from 1cm before the middle point. If the flux (magnetic field lines) is dividing at the region of point B. Then how can we know what is the amount of flux in that region where it is dividing. We should integrate the flux with respect to length in that region as the flux is varying continuously in that 2 cm range. Same Is the case for E. In that region also the magnetic field lines from both branches are getting combined and generating total flux (phi). In region E(2cm*2cm) as well, flux is changing continuously.So, how in the solution they have taken the length for calculating the reluctance by those dotted lines only.

Solution can be seen through this site:

https://holooly.com/solutions-v20/for-the-core-shown-in-fig-3-20-it-is-required-to-produce-a-flux-of-2-mwb-in-the-limb-cd-the-entire-core-has-a-rectangular-cross-section-of-2cm-x-2cm-the-magnetizing-coil-has-800-turns-the-re/

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If I understood your question correctly, the problem you are referring to is the non-uniformity of the magnetic field at the bifurcations.

In any case, Hopkinson's law and the use of reluctance is an approximate method for studying a magnetic circuit.

In your question, the lateral dimensions of the magnetic circuits are indeed too large compared to their length and it is probable that the result obtained is a poor approximation of reality.

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