Magnetic resistance of an electromagnet with varying cross section
Am I going about this the right way?


This is my method: -

Aim to find the effective permeability of the core with the air gap


Using B/H = \$\mu_e\$


B is 0.5 T as stated
H is ampere-turns per metre (5 x 200 / length)
The "per metre" part is the effective length of the flux around the core


This is really the only contentious part - how do you estimate the effective length of the core - if you ignore the longer path (to the right) you can argue that the left side route has an effective length of 44 cm. This is based on mid-point positions through the entire core including the air gap length (unknown).

So B/H = 0.00022 = \$\mu_0\mu_e\$ where \$\mu_0\$ = \$4\pi\times 10^{-7}\$

Therefore \$\mu_e\$ = 175

Then relate effective permeability to ungapped permeability


\$\dfrac{l_g}{l_e} = \dfrac{1}{\mu_e}-\dfrac{1}{\mu_i}\$

Where \$\mu_i\$ is the initial relative permeability of the core (5000)

So, the gap to length ratio = 1/181.35 and given that the length is 44 cm, this makes the gap 2.43 mm.

Of course, if I did a more detailed analysis of the core I would probably conclude that the core effective length is more like 50 cm.