I have a sphere of radius "r" , the sphere has a total charge "q" distributed equally around the surface. A point "P" is at a distance "R" from the centre of the sphere. As per the book, to find the net flux from the gaussian sphere with radius "R" , we use ∅ = E × ds × cos (theta) , here , we are assuming the electric field at every particular point in the gaussian surface to be "E" as constant. And thus ,it can be easily calculated as net ∅ = E × 4πr²(the area vector makes 0° with E). But , in the picture , I can see the Electric field can be seen as the total electric field by the all the points on the surface made by a solid angle infront of the point , and only one field is normal to the area vector while the others make different angles. So why do we compute it with taking E as constant. And if I'm correct, how can I find the net flux of that particular point and ultimately for the whole gaussian surface?

I have a sphere of radius $r$ , the sphere has a total charge $q$ distributed equally around the surface. A point "P" is at a distance $R$ from the centre of the sphere. As per the book, to find the net flux from the gaussian sphere with radius $R$, we use ∅ = E × ds × cos (theta) , here , we are assuming the electric field at every particular point in the gaussian surface to be "E" as constant. And thus ,it can be easily calculated as net ∅ = E × 4πr²(the area vector makes 0° with E). But , in the picture , I can see the Electric field can be seen as the total electric field by the all the points on the surface made by a solid angle infront of the point , and only one field is normal to the area vector while the others make different angles. So why do we compute it with taking E as constant. And if I'm correct, how can I find the net flux of that particular point and ultimately for the whole gaussian surface?

As per the book, to find the net flux from the gaussian sphere with radius "R" , we use ∅ = E × ds × cos (theta) , here , we are assuming the electric field at every particular point in the gaussian surface to be "E" as constant. And thus ,it can be easily calculated as net ∅ = E × 4πr²(the area vector makes 0° with E). But , in the picture , I can see the Electric field can be seen as the total electric field by the all the points on the surface made by a solid angle infront of the point , and only one field is normal to the area vector while the others make different angles. So why do we compute it with taking E as constant. And if I'm correct, how can I find the net flux of that particular point and ultimately for the whole gaussian surface?

I have a sphere of radius "r" , the sphere has a total charge "q" distributed equally around the surface. A point "P" is at a distance "R" from the centre of the sphere. As per the book, to find the net flux from the gaussian sphere with radius "R" , we use ∅ = E × ds × cos (theta) , here , we are assuming the electric field at every particular point in the gaussian surface to be "E" as constant. And thus ,it can be easily calculated as net ∅ = E × 4πr²(the area vector makes 0° with E). But , in the picture , I can see the Electric field can be seen as the total electric field by the all the points on the surface made by a solid angle infront of the point , and only one field is normal to the area vector while the others make different angles. So why do we compute it with taking E as constant. And if I'm correct, how can I find the net flux of that particular point and ultimately for the whole gaussian surface?