|MadSci Network: Physics|
The condition for the electric field vector to be written as the gradient of a scalar function is the curl of electric field is zero. Because if the curl of vector E is zero, the integration of vector E along different paths between two positions will be the same, therefor you can define a function of position A which is equal to the integral of vector E from a fixed position B to position A. The gradient of this function is vector E. According to Maxwell's equations, curl of E is equal to negative partial differentiation of magnetic flux density. Therefore in a region with static magnetic field, curl of E is zero, and vector E can be written as a gradient of a scalar function. After you write vector E as gradient of a scalar function, substitute it into the Gauss Law (divergence of E is equal to the charge density), then you get the Possion's equation.
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