How does the act of bringing poles closer together increase electromagnetic strength? According to my textbook, the strength of a horseshoe electromagnet can be increased in the following ways:

Increasing the amount of current flowing


Increasing the number of turns of the coil


Moving the poles closer together


Whilst I understand how the first two works, the third completely baffles me: what exactly do the authors even mean by that? It appears (at least to me) impossible to move the ends of a horseshoe magnet -- a metal! -- closer to each other as one desires. If such a doing is possible, how does the physics behind it work?
i.e. how does bringing the poles of a horseshoe electromagnet closer together increase its strength?
Image given in my TB, for reference:
Duncan, Tom. Cambridge IGCSE Physics. 3rd ed., Hodder Education, 2014.
 A: I see why that’s confusing wording, especially since the first two variations could be done to a particular magnet. But based merely on years of experience with books communicating how variables affect things (and knowledge about horseshoe magnets in particular), I can say what they meant.
They mean that, for two different horseshoe magnets, all else equal the one with closer poles (different shaped U) will be stronger. Not that we can take one particular magnet and change it in that way.
I think that was the entire question, not also that you are asking why a horseshoe magnet otherwise identical would be stronger if it had closer poles.
A: **According to Coulomb’s first law – unlike poles attract each other and like poles repel each other.
Coulomb’s second law – force between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them.**

As field strength is inversely proportional to the distance between the poles hence by decreasing distance, will increase the field strength between them. but here is a catch the field between a horse shoe magnet is homogeneous and isotopic i.e not depends on position.

but field lines those are not present between the poles are hardly or can say approximately uniform. so, if we decrease the distance between the poles the field strength (outside) will increase. here is a nice picture to understand this concept.

Horseshoe magnet with computed magnetic field lines. The two magnetic poles are in close vicinity, which concentrates the field lines and creates a strong magnetic field. hence if we take poles closer and closer, more will be concentration of field lines hence more will be strength
A: Note that the north and south poles of a horseshoe magnet are in the ends. If you were to push or bend the ends closer together, this would increase the strength of the magnetic force between the poles, since the magnetic force is inversely proportional to the square of the distance between the poles.
Mathematically, $${\bf F_m}=\frac{\mu_0}{4\pi}\frac{m_1 m_2}{r^2}{\bf \hat r}$$ where $m_1,m_2$ are the strengths of each magnetic pole, $\mu_0$ is the permeability constant and $r$ is their distance of separation.
So force example, halving their separation, will increase the force by a factor of four.
But it is also true as in Al Brown's answer, that all things being equal, the one [designed] with  closer poles (different shaped U) will be stronger.
