Re: WHAT IS THE CONDITION THAT MUST BE TRUE FOR THE ELECTRIC FILED VECTOR E TO

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.