As the magnetic field propagates does it have a momentum that can be felt by another magnetic field of the same charge? If I have a solenoid and a permanent magnet, and I placed the magnet an inch away from the solenoid, oriented in a position where the magnet would be repelled as the solenoid is turned on. Is it a correct understanding that the propagating magnetic wave from the solenoid has a momentum that would somewhat collide with the permanent magnets field, causing the permanent magnet to move away? I am thinking of a wave in a beach washing off a boat-isque analogy. And what could be the equations that can help me. Thank you
 A: Static magnetic fields by themselves have no momentum, you need an electric field and a magnetic field to have momentum.
Also, the momentum comes from the total fields. So even if you think of two magnetic fields $\vec B_1$ and $\vec B_2$ and two electric fields $\vec E_1$ and $\vec E_2$ the momentum density is $\epsilon_0\left(\vec E_1+\vec E_2\right)\times\left(\vec B_1+\vec B_2\right)$. If you expand that out you get a term like $\epsilon_0\vec E_1\times\vec B_1$ (which looks like the momentum of the first fields) and a term like $\epsilon_0\vec E_1\times\vec B_1$ (which looks like the momentum of the second fields) and terms like $\epsilon_0\vec E_1\times\vec B_2$ and $\epsilon_0\vec E_2\times\vec B_1$ (which look like the momentum of the interaction of the fields).
So if you want to talk about the momentum of the fields you need both the electric and the magnetic fields. And you can't just talk about the momentum of the fields due to this and the momentum of the fields due to that because if you do then you also need to talk about the momentum due to the interaction of the fields from the two sources (electric from one and magnetic from the other and electric from the other with the magnetic from the one).
The same thing for energy, except you can have energy with just a magnetic field or just an electric field.
Now in the region between the solenoid and the magnet momentum density (momentum per unit volume) can flow through space the net momentum that flows into a region over a time interval changes the momentum in that region by exactly the amount that flowed in over the time interval. But in the locations of charges things are different.
Inside the solenoid the field momentum can change when the electromagnetic forces act on the charges. The rate at which the charges gain or lose momentum in the say x direction is exactly equal to the rate the fields as a whole lose or gain momentum in the x direction (so equal and opposite rate because total momentum is conserved). And in the solenoid it is easier to see this exchange of momentum because the Lorentz Force Law $d\mathscr{\vec P}_\text{mech}/dt=\rho\vec E+\vec J \times\vec B$ tells us the exact rate the charges gain or lose momentum so we can tell what the fields lose or gain.
That said, we do know the momentum density, $\epsilon_0\vec E\times\vec B,$ of the fields, so we can directly compute the change if we want to.
For the permanent magnet it is a bit harder because at least some of the magnetism could be quantum. And worse, the we call it a permanent magnet but the self sustaining might be a better word because one problem is that a solenoid can actually affect a magnet not just by moving it, but it can also potentially reduce or strengthen the magnet or even change which direction the magnet's field points.
The most common formula used to model the magnet is to model it as made up of lots of little dipoles $\vec m$ each of which feels a force $\vec \nabla (\vec m\cdot \vec B)$. Snd a very simple model of that type is to have a bunch of dipoles all pointing the same direction and then have a dipole density $\vec M$ inside the magnetic. Then instead of $\rho\vec E+\vec J \times\vec B$ as a force per unit volume you use $\vec \nabla (\vec M\cdot \vec B)$ and you can integrate that to see the rate field momentum is transferred to mechanical momentum. And again, we do know the momentum density, $\epsilon_0\vec E\times\vec B,$ of the fields, so we can directly compute the change if we want to.
So I've mentioned twice that we could compute the change of $\epsilon_0\vec E\times\vec B$ directly if we wanted to. One way is if you already know the electric and magnetic fields at two times, then just compute the fields at both times and see how they change. But in your example the magnetic fields depend on the magnet so you don't already know how they change if you don't ready know how the magnet moves. But you can see how $\epsilon_0\vec E\times\vec B$ changes at a point by taking the time derivative $$\frac{\partial}{\partial t}\left(\epsilon_0\vec E\times\vec B\right)=\epsilon_0\frac{\partial\vec E}{\partial t}\times\vec B+\epsilon_0\vec E\times\frac{\partial\vec B}{\partial t}.$$
And now you might protest that you don't know how the electric and magnetic fields change in time. But strangely we do. Just like we know how things accelerate if we know the forces, then the acceleration is known and fixed and determined (well except in unusual situations that are often ignored). Similarly, we do know how the fields change because that is fixed and determined. For instance $\frac{\partial\vec B}{\partial t}=-\vec \nabla \times \vec E$ and  $\frac{\partial\vec E}{\partial t}=\frac{1}{\epsilon_0}\left( \frac{1}{\mu_0}\vec\nabla \times \vec B -\vec J\right)$. So for the solenoid we just need the fields themselves and the currents and we know how the fields changes. In fact those two formulas are exactly what you need to know how the fields change, thus how the field momentum changes.
But what about the magnet? Does it have current and if so, what is the current? If you did the simple model with a bunch of dipoles $\vec m$ all pointing in the same direction you could pretend there was a current going in a loop orthogonal to $\vec m$ and then if there is an equal density of them all the loops cancel except on the surface so you have a current in the surface and nowhere else. If there were not an equal density of dipoles then you can imagine a current like $\vec \nabla \times \vec M$. And no matter what you have a surface current with density $\vec M \times \hat n.$  So if you pretend you have those currents then you can tell his the magnetic electric fields changes and see that contribution to the change in field momentum.
Now I talked about pretending. This is because, for instance, a magnet actually depends on temperature. So in some places the dipoles point in one direct and in some places they point in others so technically there are different currents in different places so the simplest model where all the dipoles point in the same direction just isn't perfectly accurate. But you can hope that it works on average. So usually instead of talking about the actual magnetic field you average it over regions that contains many more fundamental dipoles so that the average doesn't change in radical ways.
This would be like if you saw a population density map for a city that showed colors for different regions depending on average density rather than a map that showed person sized huge densities and big empty spaces in between. The latter is more accurate, but not really what you wanted. These averages are called the macroscopic fields. And knowing them would require knowing how those dipoles affect each other and bien they respond to magnetic fields.
For that you have to know the material, and basically you need to know the dipole moment as a function of the electric field and the magnetic field and the motion of the magnet.
Without that you can't really solve it. For textbook problems they might propose very very simple models of the dipole moment as a function of the electric field and the magnetic field and the motion of the magnet. And there are situations where those simplest models can work pretty well.
