Find Direction of Flux for Bottom Hemisphere of Sphere

[quietrain]

Homework Statement

if i have a sphere just above the z=0 plane, may i know what is the inward flux for the bottom hemisphere?

namely, is it +ve or -ve? my answer says its +ve, but why? i thought the +ve flux is always pointing outwards? so shouldn't the inward flux at the bottom of the hemisphere be -ve? Example, use divergence theorem to find flux of bottom hemisphere(R=3) : assume, div F = 1. so the flux = 0.5(V) = 0.5(4/3)(pi)(3³) = 18pi.

so this +18pi is pointing up or down ??

and thus, what is the inward flux?

thanks!

 

[SammyS]

What flux?

There is no field or charge mentioned !

 

[quietrain]

erm ,if i assume div F = 1, i don't have to care about the field right? since the divergence theorem only makes use of divF ?

but in any case, the field is F = (y,-x,z) , so divF = 1 right?

i think i should rephrase my question,

if i apply the divergence theorem to the bottom half of a sphere (hemisphere:V/2) , if i get say 18pi, which way is the +18pi, which way is the -18pi pointing at?

 

[LCKurtz]

It would be helpful if you would actually state the whole problem you are working on instead of just giving us a piece of it. The statement of the divergence theorem always assumes an outward pointing normal, for what it's worth.

 

[quietrain]

Evaluate the inward flux of the vector field F = (y,-x,z) over the surface S of the
solid bounded by z = sqrt (x²+y²) and z =3

so the top surface is z=3 right? bottom is a hemisphere so i get a bowl shape?

the ans for top flux is -27∏

bottom is +18∏

integral F.n dS for top is -z(∏(3²) = -27∏ right?

but bottom if i use divergence theorem, it says volume integral of divF = flux

so divF gives 1. volume integral over the hemisphere is 0.5 ( 4/3 ∏ R³) = 18∏

now since gauss theorem assumes a outward normal, it would mean this 18∏ is pointing downwards right? so shouldn't the inward flux be -18∏now?

and hence shouldn't the answer be -27∏ - 18∏ ?

 

[LCKurtz]

Evaluate the inward flux of the vector field F = (y,-x,z) over the surface S of the
solid bounded by z = sqrt (x²+y²) and z =3

so the top surface is z=3 right? bottom is a hemisphere so i get a bowl shape?

No. The bottom surface is not a hemisphere. If you square both sides you get:

z² = x² + y²

which is not the equation of a sphere. What is it actually?

the ans for top flux is -27∏

bottom is +18∏

integral F.n dS for top is -z(∏(3²) = -27∏ right?

but bottom if i use divergence theorem, it says volume integral of divF = flux

so divF gives 1. volume integral over the hemisphere is 0.5 ( 4/3 ∏ R³) = 18∏

now since gauss theorem assumes a outward normal, it would mean this 18∏ is pointing downwards right? so shouldn't the inward flux be -18∏now?

and hence shouldn't the answer be -27∏ - 18∏ ?

When you calculate a flux integral like

$$\iint_S \vec F \cdot \hat n\, dS$$

you are calculating the flux through the surface in whichever direction you have chosen for the normal n, because that choice determines the orientation of the surface.

The divergence theorem assumes a surface enclosing a volume and the statement of the theorem uses the outward pointing normal to the surface. The answer you get is a scalar and it is not pointing in any direction. If the divergence theorem gives you a positive answer that means that the cumulative flux flow through the surface is out of the volume. If it is negative, the cumulative flow is inward. You could have a situation where the flux is inward on the lower part of the surface and outward on the upper part. These could balance out giving 0 total flux or the total flux could be outward or inward.

 

[quietrain]

No. The bottom surface is not a hemisphere. If you square both sides you get:

z² = x² + y²

which is not the equation of a sphere. What is it actually?

issn't this an equation of a sphere? i have trouble recognizing the shapes:(

in general how do i recognize shapes?

i can't tell why some times the equation is given as z = x+y, sometimes its f = x , y, z, it gets so confusing :(

When you calculate a flux integral like

$$\iint_S \vec F \cdot \hat n\, dS$$

you are calculating the flux through the surface in whichever direction you have chosen for the normal n, because that choice determines the orientation of the surface.

The divergence theorem assumes a surface enclosing a volume and the statement of the theorem uses the outward pointing normal to the surface. The answer you get is a scalar and it is not pointing in any direction. If the divergence theorem gives you a positive answer that means that the cumulative flux flow through the surface is out of the volume. If it is negative, the cumulative flow is inward. You could have a situation where the flux is inward on the lower part of the surface and outward on the upper part. These could balance out giving 0 total flux or the total flux could be outward or inward.

ah i see

 

[LCKurtz]

issn't this an equation of a sphere? i have trouble recognizing the shapes:(

in general how do i recognize shapes?

You study them until you recognize them and the standard form of their equations. Your text undoubtedly has examples showing the various quadric surfaces: spheres, ellipsoids, paraboloids, hyperboloids of 1 and two sheets, hyperbolic paraboloids and their degenerate forms, such as cones. There is no short-cut.

 

[quietrain]

ok thank you!