Q:

Confusion concerning the application of Gauss's law using symmetry arguments

Suppose we want to prove Coulomb's law from Gauss's law. The thing to do is to invoke a spherical Gaussian surface centered at a point charge, notice that there's symmetry and use that to calculate $\int \boldsymbol{E}\cdot d\boldsymbol{A}$. My concern is that while it's clear that the field must point in the radial direction, how is it that we know (using a rigorous argument) that it points radially outward for a positive charge and radially inward for a negative charge?

A:

I don't believe there is a rigorous argument involved. I believe it is purely by convention that the direction of the field is the direction of the force that a positive test charge would experience of it were placed in the field. It's really quite arbitrary. Just like the convention (in electrical engineering) that current is the flow of positive charge, even though in general current is the flow of negatively charged electrons. I suppose it doesn't matter as long as one is consistent with whatever convention is chosen.
Hope this helps. 

