Electric flux, vector or scalar?

[okami11408]

I know that the electric flux is a scalar quantity, but the concept of the Electric flux seems to confused me.

If the electric flux density is a vector quantity, how come the electric flux is a scalar quantity?

For example, I have the electric flux density: D=20i+2j.

Isn't it means that electric flux stick together in i direction is 20, in j direction is 2?

That electric flux density tells us that electric flux seems to be in i direction more than j direction, therefore it is a vector not scalar?

I may misunderstand something.

Thank you!

 

[jtbell]

If you change your coordinate system (e.g. by rotating the axes), the components of the electric flux density $\vec E$ in the new coordinate system are different than in the old one. However, the flux of $\vec E$ through a given fixed surface is the same in both coordinate systems.

 

[Studiot]

However, the flux of E through a given fixed surface is the same in both coordinate systems.

Since your first language is obviously not English you may like to know a bit of background history.

The word flux comes from flow and refers to the quantity of substance or energy or other physical quantity passing through a particular point.

So we have sunlight, heat flux, neutron flux in a reactor, the flow of fluid or fluid flux and many other instances of flux.

The flux is the sum or total over that whole area.
The flux density is the magnitude (and direction) of the flux at a particular point in the flow.

For some fluxes there is a different flow in different directions. For others such as a hosepipe, there is only one direction.

If we imagine holding up a hoop into the flow the flux is the total passing through the hoop.

If we surround our source of flow with a sphere the total amount passing through the sphere must be the same as the total amount passing through any other surrounding sphere, even if the density is distributed differently over the sphere or decreases simply because the second sphere has a larger radius.

This important fact is the basis of the inverse square law, Gauss' theorem and other theorems.

 

[okami11408]

Thanks for the replies!

However, I still confuse about what electric flux density really is.

And why it is a vector since it's a density, because from what I know density is a scalar.

Electric Flux is also a scalar as you clarified, electric flux density can't be anything but electric flux/unit area.

So, Electric flux density=(Electric flux)/(unit area)=(scalar)/(scalar)--->Scalar??

 

[Studiot]

Ok.

Let us start with the electric field.

Emanating from any electric charge we observe lines of 'electric force'.

That is we observe a physical force acting on a second charge placed at any point within this field.

This, however is a rather vague, non mathematical concept.

So to put flesh on the bones we define the electric field by this force as the vector E, whose magnitude is the value acting on the test charge and whose direction is that of the force.

Since we can assign a vector value (magnitude and direction) to E at every point around the original charge, we can imagine little arrows representing E at every point. That is E forms a vector field.

If we now offer up our small hoop of area as I decribed in post#3 we can sum all these little arrows.

Formally we do this by regarding the area as another vector, described by a vector, dA, whose direction is given by the unit normal to the centre of this area, the vector n.

We then form the scalar product of the dA vector and the E vector (which points in some different direction).
Note I have considered a small (differential area).
As in Fig1

E.dA = E dA cos(θ)

This is called the flux through the area.

If we sum this over a large area we integrate to find ∫E.dA = ∫E dA cos(θ)

We call this the total flux through a large area.

Now all this is fine if there is only vacuum betwen the two charges.
As soon as we introduce real matter (air, solids, liquids etc) we change the magnitude, but not the direction of the E vector.
This change is (observed to be) characteristic of the medium between the charges carrying the flux.

Fortunately we can calculate this effect by introducing a simple constant, characteristic of the medium concerned.

This constant is called the permittivity and is dimensionless. It is given the symbol ε.

The electric flux density vector D is defined as the Electric field vector E multiplied by this constant.

D = εE

We can then use either εE or D in calculations.

Every medium, including a vacuum has a permittivity and we often relate the permittivity of a given medium to that of a vacuum (ε₀) by the relative permittivity ε_r

ε = εSUB]0[/SUB]ε_r

The the electric flux density is a vector introduced specifically to take account of the medium in which the electric field exists.

Does this help.

 

[okami11408]

That helps a lot!

I really can't thank you enough, thanks a million!

So, the electric flux density is a vector because it is sort of electric field, but through any medium.

We need it because it can be used anywhere (in most of equation they use electric flux density instead of electric field)

It was called "density" because in terms of mathematics when multiply E with a permittivity the unit turns into C/m^2 which is the unit of density, but it is still a vector because ε is just a constant.

In physical term, it's a flow of something(charge).

And suppose I have a charge, there will be flow of electric flux density coming out of the charge (we measure them in C/m^2). And if we place another test charge near the the first charge there will be force from first charge acting on the test charge(E field).

So, If I have a charge, two effects will be occurred.

1. Flow of electric flux density from charge
2. Force(on test charge)

Do I understand the concept already?

 

[Studiot]

You are getting there there but

Flow of charge is called electric current.
You can also have something called charge density which is a true density in that it is the amount of charge in unit volume.

The electric (also called the electrostatic) field, is a 'field' of mechanical force acting between two charges.

So you need two charges for this force to exist.

The idea of flux is a (very) convenient mathematical way of describing the observation that

When any two bodies are close there is a mechanical force observable between them.

This force is proportional to the mass of each body called the gravitational force.

It is further observable that for some bodies only there is an additional force that is not accountable by gravity.

This force is proportional to the 'electric charge' on each body.

Charge is considered a fundamental property, like mass or length, time or temperature.

 

[Chestermiller]

The electric flux density vector is used to calculate the electric flux passing through any and all arbitrarily oriented cross sectional areas dA in space. Of course, for a given electric flux density vector, the electric flux passing through a given surface area will depend on how the surface area is oriented in space. The orientation of the surface area is determined by specifying the direction of its unit normal vector. If the unit normal vector to dA is pointing in the same direction as the electric flux density vector, then the electric flux is just equal to the magnitude of the electric flux density vector times the area dA. However, if the area dA is oriented perpendicular to the electric flux density vector, no electric flux will pass through dA, and the electric flux will be zero. In general, the electric flux through dA is equal to dA times the magnitude of the electric flux vector times the cosine of the angle between the normal and the electric flux vector (the dot product of the unit normal vector to dA with the unit vector in the direction of the electric flux density vector). Sometimes the unit normal vector to dA is assimilated into dA so that dA is treated as a vector (with magnitude equal to dA, and direction of the unit normal). In this case, the electric flux is equal to the dot product of the electric flux density vector and dA.