When can Gauss' theorem be applied?

[CraigH]

I'm currently reading an electromagnetism textbook and it has said that Gauss's theorem can only be applied on:

Concentric spheres
Concentric cylinders
Parallel planes

In these cases the "symmetry allows the integrals to be evaluated"

In class we only ever really use co-axle cables, micro-strip lines, parallel plates, and point charges as examples, as these all can be described by one of these 3 shapes. My question is asking about the more obscure shapes that could still technically be called one of these 3 shapes.

For example when talking about micro-strip or parallel plates the planes are always above and below each other, as shown in the picture bellow.

But can the two planes be next to each other? They would still be parallel, they are just now at the same height. For example a micro-strip with the feed and ground line both on the same side of the PCB.

 

[mfb]

It can be used everywhere where the requirements are satisfied, otherwise it would not be a theorem. This does not mean that it has to be useful everywhere, however.
A capacitor with parallel plates is probably easier to evaluate without Gauß, but this would give the same result.

 

[disknoir]

Hey

I studying em as well. The way I understand it is that the reason these symmetrical surfaces are used is because the electric field is constant in magnitude and direction everywhere on the surface. This makes the integral trivial.

Gauss' law does apply everywhere, but it is only useful in this way in when there is high symmetry.

 

[projjal]

i read somewhere that Gauss's theorem can be applied anywhere except in rotating frames

 

[PJC01]

Gauss' theorem, also known as the divergence theorem, is a fundamental concept in electromagnetism and can be applied in a variety of situations. However, as mentioned in the textbook, it is most commonly used on concentric spheres, concentric cylinders, and parallel planes due to their symmetry.

This is because the symmetry of these shapes allows for the integrals involved in the theorem to be easily evaluated. When applied to more obscure shapes, it may still technically be possible to use Gauss' theorem, but it may be more challenging to evaluate the integrals and may not provide as accurate results.

In the case of micro-strips or parallel plates where the planes are next to each other, the symmetry is still present and Gauss' theorem can be applied. However, as the planes are no longer strictly above and below each other, the integrals may be more complex and require additional calculations.

In summary, Gauss' theorem can be applied in a variety of situations, but it is most commonly used on shapes with a high degree of symmetry such as concentric spheres, cylinders, and parallel planes. For more obscure shapes, it may still be possible to apply the theorem, but it may require more complex calculations.