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no gauss low is only applicable for closed paths.

a plane sheet is not a closed path.

for applying gauss law the charge must be inside the closed loop or path...

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I'll say "yes".

Gauss' law says that the electric flux through a closed surface is proportional to

the amount of charge inside the closed surface. The shape of the surface doesn't

matter, and the shape of the charge distribution inside it doesn't matter either.

If a closed surface encloses a part of a sheet of charge, then the flux through the

surface is proportional to the amount of charge that's on the part of the sheet

inside the surface.

That doesn't bother me at all.

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