I am interested in the theory of motor design, and have built some
simple prototypes. As a first step, I would like to find out how to
optimize the coils.

Suppose a coil is wound with N turns of copper wire, of sufficient
diameter that resistance is negligible. The coil is wound around a
core of some specific material (e.g. soft iron, air) with a set of
known properties (e.g. weight, density, magnetic properties) -- you
tell me what properties are relevant. The coil is connected to a
battery with a voltage of V through a switch that can be operated
rapidly by computer control, which is used to "chop" the voltage (i.e.
cycle the switch on and off) to maintain a certain predetermined
current once that current level is reached.

The purpose of this coil is to act as an electromagnet in a motor. The
strength of the magnetic field at any moment depends on (at least) the
current, the number of turns, and the core material.

The operating procedure for this coil is to close the switch until the
current in the coil reaches the predetermined level, then chop the
voltage to maintain that current level for a specific period of time,
then open the switch to allow the current to return to zero. The total
time T between the closing of the switch and its final opening is a
constant, dependent on the RPM of the motor. During time T, the
current will rise from zero to the predetermined value for some time
T1 (switch closed), remain relatively constant at that value for T2
(chopping), and then fall to zero during T3 (switch open), where
T=T1+T2+T3. It would seem logical that T1 should equal T3. The value
of T1 is dependent on the properties of the coil, core and voltage. T2
is simply calculated as T - (T1 + T3) -- it is the time left over (if
any) after allowing for the coil rise time and fall time. At 2000 RPM,
T is about 1 millisecond.

When the core is air, it is observed that closing the switch causes
the current to almost instantly rise to the predetermined level, but
the magnetic field is relatively weak.

When the core is soft iron, it takes a significant amount of time
(e.g. hundreds of microseconds) for the current to reach the
predetermined level. I believe this is related to the inductance of
the core. However, the magnetic field is much stronger with iron than
with an air core.

When the number of turns N is increased for any given core material,
the rise time increases. With a sufficiently high N and an iron core,
the current level never reaches the predetermined value (i.e. T1 >
T/2).

There are obviously tradeoffs -- increase N or "improve" the core
material, and you increase the magnetic field for a given current
level, but you also increase the rise time which reduces the average
current level.

What is the formula or set of procedures that will enable me to model
and optimize this coil/core/voltage system? I want to be able to vary
N, V, and whatever parameters of the core material are relevant, and
calculate the change in magnetism i.e. Electromagnet Strength = f(N,
V, Core1, Core2, ...). The formula must take into account the change
in rise time caused by changes to the parameters N, V, Corei.