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HGTy
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So given any vector field V(x, y, z) in three dimensional space, is it possible to create an electric field that is described by V.
Someone told me that according to Helmholtz theorem, a vector field (with certain constrains) can be expressed as a sum of a curless field and a divergenceless field. To me since charge density is basically the divergence of the electric field, and change in B field creates curl in electric field, it should be possible to create any electric field with these two elements. The procedure would be: given V, decompose it into a curless field and a divergenceless field; then put electric charges in space such that the charge density is described by the divergenceless field at all points; similary, do the same with ∂B/∂t at each point to create curl in E field that is described by the curless field; since E field obeys the superposition principle, everything should add and the resulting E field would be V.
Now the thing is I'm not familiar with Helmholtz theorem and the conditions it requires. Can someone clarify?
Someone told me that according to Helmholtz theorem, a vector field (with certain constrains) can be expressed as a sum of a curless field and a divergenceless field. To me since charge density is basically the divergence of the electric field, and change in B field creates curl in electric field, it should be possible to create any electric field with these two elements. The procedure would be: given V, decompose it into a curless field and a divergenceless field; then put electric charges in space such that the charge density is described by the divergenceless field at all points; similary, do the same with ∂B/∂t at each point to create curl in E field that is described by the curless field; since E field obeys the superposition principle, everything should add and the resulting E field would be V.
Now the thing is I'm not familiar with Helmholtz theorem and the conditions it requires. Can someone clarify?