As the advent calendar moves into the E&M portion of the season, there are a number of possible ways to approach this. I could go with fairly specific formulae for various aspects, but that would take a while and might close out some other areas of physics. In the end, all of classical E&M comes down to four equations, known as Maxwell's equations (though other people came up with most of them), so we'll do it that way, starting with this one:
This is the first of Maxwell's equations, written in differential form, and this relates the electric field E to the density of charge in some region of space. The upside-down Delta on the left ("del" in physics terms) multiplying the E indicates the gradient operation: you take the derivative of E with respect to each of the three vector components, and add those derivatives together.
So, what does this mean, and why is it important?
The easiest way to explain what's going on on the left-hand side of this equation is to envision the electric field as a collection of little arrows at each vertex of a grid filling all of space, pointing in the direction of the force that a positive charge placed at that position would feel. The gradient operation at some point in space asks "Are there more arrows pointing toward this point, or away from this point?" If there are more arrows pointing toward that point than away from it, the gradient operation gives a negative value. If there are more arrows pointing away from the point than toward it, the gradient operation gives a positive value. If the number of arrows pointing in equals the number pointing out, the gradient operation gives zero.
What this equation tells you is that the gradient of the electric field at some point in space is related to the charge in that neighborhood. If the gradient is positive, there must be positive charge at that position. If the gradient is negative, there must be negative charge at that position. And if the gradient is zero, then that must be a point in empty space.
This seems awfully abstract, but in fact it's a very powerful tool for determining everything you need to know about electric fields. If you know the charge distribution, you can use this equation to determine the electric field everywhere in space around that charge distribution. If you know the field values at a bunch of different points, you can use them to reconstruct the charge distribution.
Nowadays, we include this as one of Maxwell's equations, but it's actually a form of Gauss's Law, formulated by the great mathematician Carl Friedrich Gauss in the early 1800's. This is a general property of vector fields, and you can write variations of Gauss's Law for other forces, such as gravity and magnetism (as we'll see tomorrow). Maxwell gets credit for this because he was in the right place at the right time, and realized how Gauss's Law for electric fields fits with the other three equations to produce a complete description of electromagnetism.
Gauss's Law, and Maxwell's equations in general, represent a sort of a shift in physics, from talking about very concrete and directly measurable quantities like force and velocity, to talking about general mathematical properties of more abstract objects, like electric and magnetic fields. This is really the point in the history of physics where the fundamental role of symmetry starts to come in, leading in the end to modern theoretical physics, in which fields are viewed as more fundamental than particles.
So, take a moment today to appreciate Gauss's Law, the first of Maxwell's equations. This isn't the last we'll hear of Gauss, though, so come back tomorrow for the next equation of the season, as we continue on toward Newton's birthday.
Gradient? I think you mean the divergence. Gradients are for scalar fields. But I like your description of the physical meaning.
I agree that divergence was the correct term. It's also worth noting that the vector form of Maxwell's Equations (the form you provided) are attributed to Oliver Heaviside, who based his work on Maxwell's original formulation. Heaviside used notations and results from vector mathematics to condense Maxwell's equations into a smaller number of equations, yielding the 4 "Maxwell's Equations".
"Divergence" is absolutely the right term. I'm an idiot. A highly distracted idiot trying to do too many things at the same time.