So the number of components doesn’t matter really: it’s just more convenient to first get A using our data on the currents j, and then we get B from A. The answer to that question is somewhat subtle, and similar to what we did for electrostatics: it’s mathematically convenient to use A, and then calculate the derivatives above to find B. B is a vector with three components, and so is A. Now, because that’s a relatively simple situation, you may wonder whether we really simplified anything with this vector potential. Note that we have no ‘time component’ because we assume the fields are static, so they do not change with time. As a start, it may be good to write all of the components of our B = ∇× A vector: Now, it’s this vector field A that is referred to as the (magnetic) vector potential, and so that’s what we want to talk about here. B = 0, then there is an A such that B = ∇× A.D = 0, then D will be the the curl of some other vector field C, so we can write: D = ∇× C.It says the following: if the divergence of a vector field, say D, is zero – so if ∇ Therefore, there’s another theorem that we can apply. Now, you can verify for yourself that the divergence of the curl of a vector field is always zero, so div ( curl A) = ∇ B = 0 equation says the divergence of B is zero, always.B = 0 equation is true, always, unlike the ∇× E = 0 expression, which is true for electrostatics only (no moving charges).Now, the two equations for magnetostatics are: ∇ So that equation sums up all of electrostatics. Φ (phi) is referred to as the electric potential. Substituting C for E, and taking into account our conventions on charge and the direction of flow, we wrote: If ∇× C = 0, then there is some Ψ for which C = ∇Ψ Therefore, we can apply the following mathematical theorem: if the curl of a vector field is zero (everywhere), then the vector field can be represented as the gradient of some scalar function: Let me recall the basics which, as usual, are just Maxwell’s equations. You’ll remember that the electrostatic field was curl-free: ∇× E = 0, everywhere. So let’s do the vector potential here, and the magnetic dipole moment in the next. Another topic I neglected so far is that of the magnetic dipole moment (as opposed to the electric dipole moment), which is an extremely important concept both in classical as well as in quantum mechanics. One of these loose ends is the (magnetic) vector potential, which we introduced in our post on gauge transformations, but then we didn’t do much with it. This and the next posts are supposed to wrap up a few loose ends on magnetism.
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