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There are two basic equations for treating static magnetic fields in matter (In the following $\mathbf{J}=0$ is assumed). The first is $\mathrm{div}\mathbf{B}=0$ which essentially means that the magnetic flux density $\mathbf{B}$ has no sources and $\mathbf{H}=\mathbf{B}-4\pi \mathbf{M}$ (here cgs-units are used) which defines $\mathbf{H}$ called magnetic ...

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The fact that the material is magnetic means that it has a magnetization, which is a source of Magnetic Field. But another interesting thing happens. Because it is made out of matter, the magnet has a different permeability $\mu$ than of vacuum, which is an obstacle to the field permeating the material. This means the resulting magnetic field will be weaker ...

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Magnetic field gradients in space can provide a force. For specifically the frog example, see https://www.ru.nl/hfml/research/levitation/diamagnetic-levitation/

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The saturation magnetisation is the maximal magnetisation that your system can exhibit. On simple lattice systems with not too complex interactions this is just the point where all your spins are aligned in some particular direction. In order to link it to temperature and magnetic moments, there are several models available. I categorise them in two ...

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I confirmed your results (2)-(4). They can be written in vector form as $$\Delta\vec{p}=\frac{\mu_0q}{2\pi b^2}\hat{v}_0\times\left[2(\vec{\mu}\cdot\hat{b})\hat{b}-\vec{\mu}\right].$$ It reduces to your result using $\hat{v}_0=\hat{x}$ and $\hat{b}=\hat{y}$. I found this by doing the whole calculation in vector form rather than using components. A ...

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