The flux through a Gaussian surface remains constant despite changes in shape, provided the net charge inside remains unchanged. This is due to the relationship between electric field strength and surface area; as the surface area increases, the electric field strength decreases proportionally. Specifically, for a point charge within a spherical Gaussian surface, the electric field decreases as 1/r² while the surface area increases as r², resulting in a constant product. Thus, the integral of the dot product of the electric field and the infinitesimal surface area remains unchanged.
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True, flux is integral of dot product of electric field and \vec{da}, that's an infinitesimal element of a surface. But if you change shape of your surface, let's say you move some part of surface away from charges, the flux through the part of your surface that you didn't touch stays the same, so if something changes, it changes on that part of your surface that you ''touched''. You moved part of your surface away from charges so area of that part is now bigger than it was, BUT also the electric field doesn't have the same value that it had on ''old'' surface. Since that part of surface is further from charges, electric field has a smaller value but their product is the same. (surface area raises, field decreases). Especially for one point charge and a spherical gaussian surface, field decreases as 1/r^2 , and surface area increases as r^2, so their product is constant.Idyia said:Why doesn't the flux through a Gaussian surface change, when the shape is changed? (while keeping the net charge inside it the same)
Flux is the dot product of electric field and surface area, so wouldn't it change if surface area is changed?
So, basically as surface area changes, the electric field changes too, so the flux remains constant... Thank you so much!Avalanche_ said:True, flux is integral of dot product of electric field and \vec{da}, that's an infinitesimal element of a surface. But if you change shape of your surface, let's say you move some part of surface away from charges, the flux through the part of your surface that you didn't touch stays the same, so if something changes, it changes on that part of your surface that you ''touched''. You moved part of your surface away from charges so area of that part is now bigger than it was, BUT also the electric field doesn't have the same value that it had on ''old'' surface. Since that part of surface is further from charges, electric field has a smaller value but their product is the same. (surface area raises, field decreases). Especially for one point charge and a spherical gaussian surface, field decreases as 1/r^2 , and surface area increases as r^2, so their product is constant.