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I want to know why every Gauss's law question starts with an assumption that a rod, cylinder, plane, etc. is very/infinitely long.

Examples:

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  • $\begingroup$ Consider to give one or two examples, with references. The question totally lacks context. $\endgroup$ Apr 23 at 11:26
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    $\begingroup$ I don't know the true context, but I suppose it's due to the simplification of boundary conditions, i.e. otherwise field lines at edges becomes non-trivial. $\endgroup$ Apr 23 at 11:34
  • $\begingroup$ @TobiasFünke I'm sorry. I added examples. $\endgroup$ Apr 23 at 11:39
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    $\begingroup$ Don't use images for text and equations. Use MathJax instead, i.e. type the relevant passages. And give a references, e.g. book title, edition, page, equation or problem number(s)... $\endgroup$ Apr 23 at 11:56
  • $\begingroup$ Voting to reopen. OP is asking a conceptual question and not how to do the problem(s). $\endgroup$ Apr 23 at 16:24

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The main use of Gauss's law (as well as Ampere's law) is in configurations of high symmetry. The key argument is that the electric field depends only on one coordinate, so that it is constant on a Gaussian surface and can be pulled outside the integral. In the case of cylindrical symmetry, it depends only on the radial coordinate $s$ and not on the position along the cylinder $z$ and the azimuthal angle $\phi$. In the case of planar symmetry, it depends only on the perpendicular distance from the plane $z$ and not on the position parallel to the plane, $(x,y)$.

This argument is only possible if the charge configuration remains the same regardless of the other coordinates. A finite cylinder doesn't look the same from every $z$ but an infinitely-long cylinder does. Similarly, a finite plane doesn't look the same from every $(x,y)$ but an infinite plane does1. That's why we cannot eliminate dependence on the other coordinates in the finite case.

However, it is often still a good approximation when the length/size of the cylinder/plane is much larger than the distance of the point of consideration to the cylinder/plane.

Note that there is one commonly used Gaussian surface that is not infinite and that is the sphere.


1In fact, an infinite plane looks the same from not only any $(x,y)$ but also any perpendicular distance from it. However, the argument for that is slightly more subtle.

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To successfully use Gauss's law to find the magnitude of the electric field, we need to be able to say something about its magnitude and/or direction on some surface. This is what allows us to convert from the flux integral $\int \vec{E} \cdot d\vec{A}$ to an expression of the form $|\vec{E}| A'$ (where $A'$ is the area of some or all of the Gaussian surface.)

Without these symmetry assumptions, Gauss's Law simply can't be used to find the field. All we would know is that the total flux through the surface is equal to some number; but since we would have no idea how this flux was distributed over the surface, or about the orientation of the electric field on that surface, we would be unable to say anything meaningful about the magnitude and direction of the electric field at any particular point.

For a "truly" infinite cylinder or rod, we can argue from the symmetry of the situation that the electric field should point directly away from the rod at all points, and that the magnitude of the field should depend only on the distance from the symmetry axis. Similarly, for a "truly" infinite plane or slab of charge, the electric field should point directly away from the plane at all points, and should only depend on the coordinate perpendicular to the slab. Of course, "truly" infinite charge distributions don't/can't exist; but if a charge configuration is "very large" or "very long" then we can use the infinite case as a good approximation.

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As others have pointed out, it's to have the required symmetry needed in order to pull $E$ out of the integral.

The analogy I like to give is the following. Let's say I told you I have ten numbers that add up to 100. Can you tell me the numbers that I used? Of course not; there are infinitely many ways for ten numbers to add up to 100. But, what if I told you that all of the numbers are equal? Then you can definitely say each number is equal to 10.

We need specific symmetries in order to use Gauss's law in integral form to determine the Electric field of a charge configuration. More specifically, you have to be able to define Gaussian surfaces where $\mathbf E\cdot\text d\mathbf a$ is the same value on each side of the surface such that

$$\iint \mathbf E\cdot \text d\mathbf a=\iint E\ \text da=E\iint\text da=EA$$

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Why are all the Gaussian surfaces very long or infinite?

The immediate counter-example is a spherical surface around a point charge or a charged sphere.

As the others have already pointed out, the choice of the surface has to do with the high symmetry of the problem, which reduces the surface integral to multiplying a constant field (component) by the surface area:
(image source) enter image description here

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If you want to calculate the electric/magnetic field outside/inside a finite cylinder, you're gonna have to use Bessel functions. To be fair outside/inside a sphere is the spherical harmonics functions and not just a simple $1/r$, but it's the first order of the explansion so it's ok.

You get these from solving the Laplace equation in cylindrical/spherical coordinates. Dont dig into this rabbit hole if you haven't learned about Fourier series and transforms first.

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