# B Decoding flux of a vector field

1. Sep 19, 2016

### manasi bandhaokar

i was going through Gauss law and the chapter started with flux of a vector field.i understand it mathematically but not physically,
i have been reading on the net and most common explanation is that it is the amt of "something"(anything) crossing a given surface.fine till here.then i read that total flux through any closed surface is zero.i again understand it mathematically - positive and negative flux for entering and leaving but not physically.if "something" IS flowing through the surface and flux is amt of "something" flowing through the surface the why does total flux which i interpret as the total amt of "something" crossing the surface zero?

Last edited by a moderator: Sep 21, 2016
2. Sep 19, 2016

### Orodruin

Staff Emeritus
This is not true. It is only true for a source free field.

It is then the net sum of what flows into the volume that is zero, i.e., what flows in minus what flows out.

3. Sep 19, 2016

### Staff: Mentor

If there is a net flux in or out, there can also be "something" accumulating or depleting inside the volume. There doesn't need to be a source within the volume for this to happen.

4. Sep 19, 2016

### Orodruin

Staff Emeritus
True, I had the static situation in mind when I wrote that.

5. Sep 19, 2016

### manasi bandhaokar

Really sorry but i still dont get.flux is a measure of no of field lines crossing the given surface.if say 5 field lines flow in then 5 will exit.but here are these 5 field lines crossing the surface .then why is the flux 0?

6. Sep 19, 2016

### Orodruin

Staff Emeritus
Becuse everything going in comes out.

7. Sep 19, 2016

### manasi bandhaokar

???
Sorry

8. Sep 19, 2016

### Orodruin

Staff Emeritus
Flux is not a measure of the number of field lines, it is a measure of how much flows through a surface based on a local current density.

If you have a closed surface and a divergence free field, there may very well be parts of the surface where stuff flows in. However, this will be exactly cancelled by a flow out of the region at a different place on the surface.

9. Sep 21, 2016

### manasi bandhaokar

Flux is not a measure of field lines???!!! but my physics teacher said so.and if its not that what is it?can u please explain in simpler terms.

10. Sep 21, 2016

### Orodruin

Staff Emeritus
No. It is only true in some very particular cases. You can have different vector fields with the same field lines (they are just the integral curves of the vector field). Do realise that what your physics teacher says will generally be aimed at you understanding things (often pictorially) in a particular regime or for a particular case.

Flux is a measure of how much "stuff" flows through a surface given a particular current field. For example, if you have a fluid, the mass current will be given by $\vec J = \rho \vec v$, where $\rho$ is the density and $\vec v$ the velocity field. The flux through a surface will tell you how much mass flows through that surface per time unit.

11. Sep 21, 2016

### A.T.

12. Sep 21, 2016

### manasi bandhaokar

ok
okay.you say flux tells us how much "stuff " flows through a surface given a current field.so if the flux through a closed surface is 0 then it should imply that nothing is flowing through it.right?but isnt there something flowing through it?

13. Sep 21, 2016

### vanhees71

Field lines are just a picture to show the direction of the field at any point in space, i.e., the tangent vectors to these lines point in direction of the field. The more or less complete information is contained in pictures, where they draw little arrows at each point with the length proportional to the magnitude of the field.

To get a physical picture about the flow of a vector field, we take up Orodruin's example from #10. At the same time it defines what are flux densities, i.e., vectors like the mass current density Orodruin mentions above or the electric current density (note that these quantities are always densities, in the precise sense to be derived now).

To get the idea about flux right, it is first important to remember, how surfaces are described. A surface by definition in our context is a two-dimensional smooth subset of Euclidean 3D space. You can describe it (or at least some part of it) by two parameters $(u,v)$ as a function $\vec{x}(u,v)$. To get an idea about the orientation of this surface in space, we define the surface-normal vectors,
$$\mathrm{d}^2 \vec{f} = \mathrm{d} u \mathrm{d} v \partial_u \vec{x} \times \partial_v \vec{x}.$$
Geometrically the magnitude of this vector is the area of the little parallelogram spanned by the infinitesimal tangent vectors defined by the "coordinate lines" $v=\text{const}$ and $u=\text{const}$, and the direction is perpendicular to both of these tangent vectors (remember the geometrical meaning of the vector product). Also note that there are always two possible orientations of each of these surface vectors. You can switch from one to the other by exchanging $u$ and $v$. For the following it is not important, but one has to be aware that choosing the one or the other orientation is arbitrary and changes the sign of the surface intgrals used now to define flux.

Suppose now you have a such defined surface with an orientation defined by the order of $u$ and $v$. Then consider a fluid of mass density $\rho(\vec{x})$, and we ask the question, how much mass goes through the surface in an infinitesimal time $\mathrm{d} t$. To that end we need also the flow-velocity field, i.e., $\vec{v}(\vec{x})$, which describes the velocity of the fluid at each point in space (or at least along the surface). Now in a little time $\mathrm{d} t$ a fluid element at $\vec{x}$ moves a little distance $\mathrm{d} t \vec{v}(\vec{x})$. The volume of the little cylinder of fluid built by the surface element $\mathrm{d} \vec{f}$ flowing through this surface element is given by $\mathrm{d} V=\mathrm{d} t \mathrm{d}^2 \vec{f} \cdot \vec{v}.$
Thus in this time $\mathrm{d} t$ a mass of
$$\mathrm{d} m=\rho \mathrm{d} V = \mathrm{d} t \mathrm{d}^2 \vec{f} \cdot \rho(\vec{x}) \mathrm{v}(\vec{x})$$
runs through the surface element. Summing over all surface elements means to integrate over the surface, and you get the mass per unit time running through the surface by the flux
$$\Phi=\int_F \mathrm{d}^2 \vec{f} \cdot \vec{j}(\vec{x}), \quad \vec{j}(\vec{x})=\rho(\vec{x}) \vec{v}(\vec{x}).$$
The sign of the flux just tells you if the net flow per unit time is in (positive) or against (negative) the direction of the choice of the orientation of the surface-normal vectors. So no matter how you choose the orientation of these vectors, the physical meaning is the same. One must only carefully specify the choice of orientation to make sense of the direction of the flow.

14. Sep 21, 2016

### manasi bandhaokar

i understand that my inability to understand this might be irritable.i am sorry for that.

15. Sep 21, 2016

### Orodruin

Staff Emeritus
No, if the flux through a surface is zero, it means that as much is flowing through the surface in one direction as is flowing in the other direction. It does not mean that the flux is zero through every part of the surface.

16. Sep 21, 2016

### manasi bandhaokar

i am sorry but all this is way beyond my understanding am only in 12 std.

17. Sep 21, 2016

### Staff: Mentor

The lines count as negative when they flow in (enter the surface). They also count as positive when they flow out (exit the surface). -5 + 5 = 0.

18. Sep 22, 2016

### cnh1995

Ok. There are many physics experts who prefer not to use the concept of "field lines" for visualizing a vector field. But for junior college level physics (at least in India), I think almost every book uses this concept to explain flux.

In this picture, you can see magnetic field lines are passing through a circular surface (which is tilted at various angles). Say magnetic field B is constant everywhere on the circular surface. Now, magnetic flux will be defined as Φ=B*A (it's actually the surface integral of B⋅dA, but since B is constant everywhere, Φ=BA).
Keeping the area same, if you want to change the flux, you'll need to change the magnitude of B.
This flux is pictorially represented using "field lines". Number of field lines passing through the surface will be directly proportional to the flux. It is just a convenient pictorial description of the term "flux". More lines-more flux, fewer lines-smaller flux. The surface area should be "normal" to the field lines. Hence, when tilted, area vector changes.
Consider a hollow metal sphere placed in an external electric field. This is an example of a closed surface. Electric field lines will enter the sphere and all the entered lines will leave the sphere. This means, net flux through the surface area of the sphere is 0.

Last edited: Sep 22, 2016
19. Sep 22, 2016

### robphy

Possibly enlightening to play with...
http://www.glowscript.org/#/user/ma...older/matterandinteractions/program/13-fields
but you might need some guidance as to what to do.

Select Measurement Type.... Gauss Law.
With your mouse, try to draw a closed loop (it's actually meant to be a closed surface... "Gaussian surface").... you won't see anything as you draw it.
[It will actually close the loop/surface for you.]
Now drag a red charge (it's a positive unit of charge)... and move it around [and the Gaussian surface will be revealed]...
Move slowly so you can see the pattern of flux-calculations through "tiles" of the closed surface.
"Q" refers to the net charge inside that Gaussian surface.
Move the charge through the surface.
Keep the charge far outside.
Next, drag a blue charge (it's a negative unit of charge) and do the same.
One at a time, slowly put both charges inside.
Set up a "dipole".. with one charge inside near one end of the surface, and the other charge inside at the opposite end [to emphasize the flux].
Put one charge in the middle... then bring the other charge on top of it.

As a variant, start over. Introduce a charge. Then draw Gaussian surfaces.
Try two nested surfaces. Try two separate disjoint surfaces.

[While a given field line shows the direction of the electric field...
the density of field lines shows the strength of the electric field.... how many units of flux per unit area.]

20. Sep 26, 2016

### manasi bandhaokar

this entry - exit thing is ok .my point is - field lines enter , cross the surface,and then leave so field lines always exist inside the surface so how can flux - measure of no of field lines - be 0?is it that i am getting the flux concept wrong?if yes then what exactly it is?