Theoretical physics is the field that develops theories about how nature operates. It is fundamentally physics, in that the ultimate goal is to describe reality. It is informed by experiment, and at the same time it extends the results of experiments, making predictions about what has not been physically tested. This is accomplished using the language of mathematics, and often the demands of theoretical physicists force mathematicians to extend this language in new directions, but it is not concerned with developing the language of math. Theoretical physicists are, among other things, physicists who are very well-versed in math (which is not to say other physicists are not - please don't hurt me).
Mathematical physics, on the other hand, is a branch of mathematics. It explores relations between abstract concepts, proves certain results contingent upon certain hypotheses, and establishes an interlinked set of tools that can be used to study anything that happens to match the relations and hypotheses on hand. This branch in particular is motivated by the theories used in physics. It may seek to prove certain truths that were simply assumed by physicists, or carefully delineate the conditions under which certain theories hold, or even provide generally applicable tools to physicists, who can in turn apply them to nature. Mathematical physicists are mathematicians who are intrigued/inspired by physics.
One could say that mathematical physics is concerned with the internal, logical consistency of physical theories, while theoretical physics is concerned with finding the right model to describe the world around us. Very roughly, one might diagram these things as shown below.
$$ \text{Mathematical physics} \Longleftrightarrow \text{Theoretical physics} \Longleftrightarrow \text{Experimental physics} $$
An electromagnet is a made coil associated with a ferromagnetic core. This way, the strength of the magnet is controlled by the input current.
A solenoid is a simple shape used in magnetostatics or magnetics. Like the plane or the sphere in electrostatics, the 1-turn coil in magnetostatics, its study is interesting because the calculus of the magnetic field inside is doable. Moreover, the solenoid produces a pretty uniform field inside, if you are neglecting edge effects.
So you could say that the solenoid is interesting because of the uniform magnetic flux density inside, and the electromagnet because of the non uniform magnetic flux density outside (an electromagnet may be done with a solenoid).
Best Answer
A perturbation is a small change (usually deterministic and known), while a fluctuation is a (not necessarily small) random perturbation with mean zero (and therefore either unknown or unrepeatable).
Usually one talks about a perturbation in the context of perturbation theory. Perturbation theory is used to study a system that is slightly different from a nice system where you can do calculations easily; the difference from the nice system is the perturbation.
On the other hand, one talks about fluctuations when deriving results for the mean of certain quantities; the fluctuations are the deviations from the mean. Another common word for fluctuations is noise (used in the singular only). In realizations of stochastic processes, fluctuations are visible as contributions of large but irregular frequency, whereas a perturbation would be in this context a small change of a parameter of the system or the forcing term.