On the electromagnet setup pictured below, only the core part is magnetic (when powered, it attracts a piece of steel) but not the 2 "wings" outside that are supposed to channel the magnetic flux.
Why is that?
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The strength of an electromagnet's field will typically be strongest on the inside of the coil (talking about a normal electromagnets). This is true for the geometry of the magnet you've shown in the question.
When you run a current through the coil, magnetic fields (perhaps not the best way to phrase this) are produced along the path of the coil that are always perpendicular to the coil. You can see an illustration of this below in an image on Wikipedia's page on electromagnets:
What you can observe is that in the center of the coil the magnetic fields produced along the path of the coil tend to converge (geometry dependent). On the outside of the coil these fields diverge from each other. It should be easy to see that the field strength inside the coil will be stronger than the outside.
The configuration of your electromagnet is a little confusing. The coil is oriented perpendicular to the long axis of the core, which means the "wings" as you call them don't really get a strong field as they are outside the coil. If the coil was rotated 90 degrees so that the core was aligned with the long axis of the coil then I would agree that the field should be strong throughout the whole core.
I'm no physicist, this is the easiest way I can explain it. If someone else can produce a more technical answer you'd benefit more from that.