17. We're ready now for the first of Maxwell's equations: Gauss's law. This law says that the divergence of the electric field is proportional to the charge density. What does the law say about the flux of the electric field through an enclosed volume?
From Quiz Maxwell's Equations
Answer:
It's proportional to the charge inside
These are really just two equivalent ways of stating Gauss's law. Start with the differential form of the law: the divergence of the electric field E is equal to the charge density, rho, times a constant. Now let's integrate both sides over some volume. (Integration is a mathematical method for adding up all the values of an expression in some region, in this case a volume of space.) A charge density has units of charge per volume, so when you integrate it over a volume, you end up with the total charge. Meanwhile, there's a theorem of calculus that tells us how to integrate the divergence of E over a volume: it's the same as integrating E over the surface that encloses the volume, and we know that's the definition of the flux. So, there we have it: the flux of the electric field through a closed surface (that is, a surface that goes all the way around a volume) is equal to the charge inside, times a constant.
What's the constant? This depends on what units you're using and what material you're studying. In a vacuum and in SI units, which include such favorites as meters, Volts, and Amperes, the constant of proportionality here is one divided by epsilon0, the permittivity of free space. This has been measured to be 8.85 million-millionths Coulomb-squared per Newton per meter-squared.
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