Let's say we have current density $J_i$ on a discretized grid with $(N_x \times N_y \times N_z)$ points. What is the best procedure to compute the induced magnetic field $(B_i)$ from the current density vector, $J_k = \epsilon_{ijk} \partial_i B_j $? I am more concerned with the numerical procedure of the solution.
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2$\begingroup$ I don't know why this was closed, it's asking for advice on a numerical method for a particular physics problem, which is one of the comp-phys topics we do address. $\endgroup$– Kyle KanosCommented Jan 25, 2023 at 18:55
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$\begingroup$ Actually, I misread the question (thinking it was how to compute $J$ from $B$). You are unfortunately unable to uniquely determine $B$ in this case as you can add any arbitrary curl-free vector field to the solution and obtain the same current density. $\endgroup$– Kyle KanosCommented Jan 26, 2023 at 0:48
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$\begingroup$ Hi Kyle! Thank you for your comment. I do understand that B is not uniquely determined in this case. However, even if we fix that by adding an arbitrary curl-free vector (analogous to setting the gauge), the numerical procedure to uncurl the RHS still remains of interest to me. Something along the lines of extending the Biot-Savart law to moving fluids. I was hoping somebody might have already done something like that, but to no avail apparently $\endgroup$– myreshCommented Jan 26, 2023 at 14:44
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$\begingroup$ My experience is in numerical MHD simulations, but I've only ever needed to compute $\mathbf{J}$ from $\mathbf{B}$ for resistive MHD (i.e., you write Faraday's law using $\partial_t\mathbf{B}=\nabla\times\mathbf{v}\times\mathbf{B}+\nabla\times\eta\mathbf{J}$ where the last term ends up being $\sim\nabla^2\mathbf{B}$). $\endgroup$– Kyle KanosCommented Jan 26, 2023 at 16:14
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1 Answer
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As long as electric field throughout the system is a potential field (gradient of some function of position), the Biot-Savart law (which gives magnetic field as integral of a certain function of current density and position all over the system) applies. This includes the cases where electric potential changes in time, but induced field is negligible.
However, this only gives you contribution to total magnetic field due to that current density, it doesn't give you contribution due to other sources, such as those implied by some specific boundary condition like "field vanishes on one side" or "field has zero normal component on some face of the cuboid region". Including such boundary conditions can be done if we go back to the Maxwell equation
$$ \nabla \times \mathbf B = \mu_0 \mathbf j + \mu_0\epsilon_0\frac{\partial \mathbf E}{\partial t} $$ and apply the Helmholtz decomposition theorem to express $\mathbf B$ everywhere in terms of $\mu_0 \mathbf j + \mu_0\epsilon_0\frac{\partial \mathbf E}{\partial t}$.
https://en.wikipedia.org/wiki/Helmholtz_decomposition
See the formula for vector field $\mathbf A$ in terms of field $\mathbf F$; this can be directly used in the same way to get the vector field $\mathbf B$ in terms of the field $\mu_0 \mathbf j + \mu_0\epsilon_0\frac{\partial \mathbf E}{\partial t}$. However, as you can see, knowledge of current density is not enough, one needs to know rate of change of electric field everywhere as well. Only in the special case electric field is a potential field (like the electric field inside a capacitor that is slowly charged), the contribution due to the electric term vanishes and it is enough to know current density everywhere, and the result reduces to the Biot-Savart formula.
I can't give you advice on numerical method. I would try basic Riemann integration. It's probably better to ask elsewhere on SE on numerical methods in physics, e.g. https://scicomp.stackexchange.com/ .
answered Jan 27, 2023 at 1:36