I have just finished watching Khan Academy tutorial on divergence. Therein he uses fluid velocity to represent vector fields. Positive divergence at a point represents source of mass at that point.

Comparing it with the expression $\vec \nabla. \vec{E}$, it is evident that electric field is the analog to fluid velocity.

However it seems to me that we cannot make charge as the analog of fluid mass. This is because over the whole vector field domain, we have fluid mass present. But we don't have electric field present over the whole electric field domain.

So in electrostatics, what shall we make the analog of fluid mass?

  • $\begingroup$ There would usually be electric field present over the whole space. Instead, there would likely be no electric charge at most points. Perhaps there is a typo? $\endgroup$ – edm Mar 17 '18 at 8:37

Consider the simple example of a constant-density fluid flow. In other words, the mass flux through a surface is proportional to the velocity field. This means that for such a flow, we can stop thinking in terms of mass and think simply in terms of velocity - flux is simply the integral of the normal component of the velocity field over a surface. This subtle shift of conception means that we now think of points of positive divergence as sources of flux (of the velocity field) rather than sources of mass.

Analogously, charges would be the sources of flux (of the electric field). This would be a more perfect analogy.

Btw, there is another situation where mass does function analogous to electric charge. That would be while considering the gravitational field (with masses as sources of flux of that field) and not fluid flow.

Gravitational field : Mass ~ Electric field : Charge

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