Ideal dimensions for an electromagnet?

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For an air-core solenoid, the geometry that gives the greatest magnetic field for a given quantity of conductor is to have the outer diameter of the coil to be approximately twice the inner diameter, and the cross-section of the coil to be approximately square. These dimensions give what is known as a Brooks coil.

(Image from https://coil32.net/multi-layer-coil.html)

For a ferro-magnetic core, the greatest magnetic field for a given amount of core material is a toroidal geometry. HOWEVER, such a geometry is useless as an electromagnet because, apart from leakage, the magnetic field is confined to within the core. Such a geometry provides little useful field outside of the core.

Thus, an electromagnet with a ferro-magnetic core needs an air-gap. The larger distance across the air gap, the weaker the magnetic field (other things being equal). However, the air gap needs to be large enough to exert the desired force on whatever the electro-magnet is used for. Thus, there is no one-size-fits-all solution.

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If the purpose of the electromagnet is to hold a flat plate of ferro-magnetic material, I strongly believe that a "potted coil" would be the best. That is, the core consists of a cylinder of ferro-magnetic material with a circular channel cut into one face, with the coil nested in the channel. Something like this

but open on the top, and the top of the coil flush with the face with the channel in it. [The image shows a potting core that consists of two halves that are supposed to be brought together to close the magnetic circuit. That is NOT what I mean. I mean using only one of the two core halves, and only half the coil.]

Be aware that many cores of this geometry use ferrite rather than "soft iron". These two materials have significantly different magnetic properties. Ferrite has lower permeability, so for an electromagnet, "soft iron" is much better. You may not be able to easily find a soft iron core in that geometry, so, for a "do-it-yourself" electromagnet, an "E" core from an "EI" transformer will probably perform much better, even though the geometry is not "ideal". Here is what I mean by the "E" core of an "EI" transformer.

(Image from https://www.delatsch.com/product/e-i-transformer-laminations/)

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