Good explantion of Gauss' Law but I have a question

[rockyshephear]

Here's the good explanation.
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A surface is an abstract mathematical tool that we can use to explore various phenomena. What a surface is, is a shell around a region of space. Consider a common inflatable ball, such as a soccer ball, or a basketball. The "ball" itself is just a rubberized shell around a pocket of air.

Like a ball, a surface is just a shape in space that encloses a certain volume. Like a ball, the surface cannot have any gaps in it (if a ball has gaps, air will escape, and the ball will deflate).

If the surface we have is a closed one, so it has no gaps, then you can imagine that in a river the same amount of water that enters the surface must also exit the surface. The only time that more water can be coming out of a surface is if there is a hose inside the surface. The only time more water can be entering the surface is if there is a drain inside the surface.

Or if we go back to Electric Field, the only time the flux in a surface is not zero, is when there is a charge inside that surface.


[Gauss' Law for Electric Fields (zero charge)]

\oint_S{\mathbf{E}\cdot d\mathbf{a}} = 0 ---

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The anaology is losing me a bit. I'm imaginging this surface (like a balloon submerged in water) I can see the surface as being the skin of the balloon. Water outside the balloon can somehow magically entere the balloon even though it's a closed surface (not going there). But the water that is reportedly entering the balloon that they say is equal to that entering is ending up INSIDE THE BALLOON. How is that supposed to be water leaving the surface?
I'm wondering why the suface has to be spherical. Maybe a sheet of paper is better for the analogy. Water enters one side and leaves the other. Then Gauss' Law states that the difference between the the volume of water entering the surface and leaving the surface is zero because there is no source or sink, no charge.

Or maybe they are saying without a 'charge' inside the balloon, water enters the balloon, fills the balloon and exits the other side of the balloon. I can buy that. Is that what they are saying?

 

[turin]

Think of a fine closed mesh, or closed netting, as the closed surface. Then, water can move in and out freely, and the surface can be any shape that you want. The purpose of the surface is not to keep things in or out, so you don't need an impermeable surface like a baloon. The purpose of the surface is simply to define some nonphysical boundary, and you don't want the nonphysical object to have any effect on the physical stuff (the water).

 

[charng]

I appreciate your question and your attempt to understand the concept of Gauss' Law. The analogy of a surface being like a balloon or a ball is meant to help visualize the concept, but it is not a perfect representation. The key idea of Gauss' Law is that the flux (or flow) of something through a closed surface is equal to the amount of that thing inside the surface. In the analogy, the "thing" is water, but in reality it could be anything that is being measured, such as electric charge or magnetic fields.

To address your question, let's use the analogy of a sheet of paper instead of a balloon. Imagine a sheet of paper submerged in water, with water flowing in through one side and out the other. The amount of water flowing in must be equal to the amount flowing out, otherwise the water level would change. In the same way, the flux of an electric field through a closed surface must be equal to the amount of charge inside the surface. Without a charge inside, the flux is zero.

The shape of the surface does not have to be spherical, it can be any closed shape. The reason a spherical shape is often used in Gauss' Law is because of its symmetry. If the charge is located at the center of the sphere, the electric field will be the same at every point on the surface, making the calculations easier. But this is just a simplification and does not change the fundamental concept of Gauss' Law.

I hope this helps clarify the concept for you. It's important to remember that analogies can be useful, but they are not always perfect representations of the actual concept. It's always a good idea to think critically and ask questions to fully understand a scientific concept.