What would the magnetic field of two (practically identical) static cylindrical permanent magnets stuck together (called compound magnets) look like?

Question

I am doing a science fair project on the Meissner effect and to achieve stability I had to put two cylindrical magnets next to each other (not on top, end to end) because I was levitating a superconductor above a magnet instead of a magnet above a superconductor. If I had been able to use just one magnet the field calculations seem rather straightforward (albeit complicated) but after considerable research, I found almost nothing about how to calculate the magnetic of two cylindrical magnets put end to end to each other forming what amounts to a magnet array. I found one article that described two cylindrical magnets stuck together in this manner as a compound magnet but they didn't do any calculations of the magnetic field.

I found this equation:

On Wikipedia on the article "Force Between Magnets" but these equations are for magnets on top of each other. I also found this website https://www.supermagnete.de/eng/faq/How-do-you-calculate-the-magnetic-flux-density, which details the different equations that can be used to calculate individual B fields (magnetic fields), but I am not sure how to apply the equations to multiple magnets.

A basic diagram of these magnets:

The magnets I used were the exact same size, this is just a basic diagram of how I arranged my magnets.

My question is if anyone would be able to help me to understand how to go about calculating the magnetic field of these two magnets? I am a high school senior and while I have an advanced math background these equations are very difficult for me to understand. My mom, who is a college math professor, is trying to help me through the mathematics but she doesn't understand the physics behind interacting magnetic fields.

Answer 1

Magnetic fields obey something called superposition. This means that if Magnet #1 creates a field $\vec{B}_1$ at a certain point in space without Magnet #2 present, and Magnet #2 creates a field $\vec{B}_2$ at that same point in space without Magnet #1 present, then the total magnetic field at that point in space when both magnets are present is simply $$ \vec{B}_\text{total} = \vec{B}_1 + \vec{B}_2. $$ Note that this is a vector sum, i.e., you would need to decompose $\vec{B}_1$ and $\vec{B}_2$ into components to actually do this addition.

This assumes that the presence of each magnet doesn't actually change the strength of the other one or anything like that. For the case of strong ferromagnetic materials, this may not be precisely true, but it's probably a pretty good approximation. I would treat this assumption as a working model for how your magnetic field behaves; it's possible that you'll need to refine this model as your project progresses and you collect more data, but that's how the scientific process is supposed to work.