Maxwell`s First Equation: Gauss’s Law (Video explanation below ↓↓↓↓)

Imagine a balloon filled with positive charges — not air, but charges — spread throughout the inside. These charges push electric field lines outward, just like gas would push outward on the walls of the balloon. Now imagine you’re measuring how hard that gas is pushing against the balloon’s surface — that’s what the electric field is doing. You’re not measuring the charges directly — you’re measuring how much electric field is leaking out of the surface due to the charges inside, just like how air inside the balloon tends to push against the wall trying to escape. But there’s an important difference: the air stays trapped inside the balloon (ignoring small leak), while the electric field generated by each charge actually leaks outside. If the balloon contains only positive charges, it’s like high pressure — field lines radiate outward (Figure below). The figure below shows a small patch to imagine how charges inside the balloon contributing to the outward field. If it contains only negative charges, it’s like a vacuum pulling inward — field lines point inward, as depicted in Figure B. If it has equal positives and negatives, the pressures cancel out — there’s no net push, and the electric field flux through the surface is zero.

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What are we really measuring here? We can't directly count the charges hidden inside the balloon just by looking at its surface. Instead, Gauss’s Law provides a powerful method to determine how much net charge is enclosed, by measuring the total "flow" of electric field through the closed surface. The fundamental insight is that the amount of electric field leaking outward (the total flux) is directly proportional to the net charge trapped inside.

This relationship is captured mathematically by Gauss’s Law:

Just like air trapped inside the balloon, Gauss’s Law tells us how much electric field is flowing out of a closed surface, and that flow is directly proportional to how much net charge is trapped inside. The law clearly says, "closed surface," and the balloon is a good example of that because, just like no air can escape the balloon, no charge can escape the surface. Let’s go from the left side of the equation to the right. We use a surface integral, and that little circle on the integral sign indicates that it is a closed surface. Why do we use an integral, not some other function? Because it’s easier — and smarter — to look at a small patch of the balloon’s surface rather than the whole thing at once. The integral sign represents the process of summing up all those tiny pushes over the whole balloon. It’s a closed integral — like a sealed balloon — because we’re enclosing all the charges inside. Nothing escapes. This method also works no matter what shape your surface is — sphere, cube, peanut — the math still holds.

We draw arrows pointing outward from that patch — those arrows represent the electric field vectors. They show us both the direction and the strength of the field at that location. As mentioned above, if there are more negative charges, the field arrows point inward toward the balloon, and the total electric field would be negative — or vice versa. That’s how we measure the electric field at that specific region: by focusing on a tiny part and seeing how much "push" or "pull" it feels. At each small patch dA, we consider the electric field vector and calculate its contribution to the outward "push" (or inward "pull"). The dot product measures how much of the electric field is actually passing through that specific area. By summing over the whole surface, the integral gives us the total electric field flux.

The terms themselves have clear physical meaning E is the electric field vector, representing the force per unit charge at each point—our "push" or "pull." qinside​ is the net charge trapped within the closed surface. And ε0 the permittivity of free space, tells us how easily electric fields can form and spread in vacuum. You can think of permittivity as a kind of "resistance" for electric fields (not for currents): higher permittivity means it is harder for field lines to spread.

In summary, Gauss’s Law provides a remarkably elegant way to connect the local behavior of the electric field at the surface to the total amount of charge hidden inside. By integrating the field over a closed surface, we learn how much charge is enclosed—without ever having to see or touch those charges directly.

Please, watch the video explaining Maxwell`s First equation: https://youtu.be/WJXUNvCRoSw