I Another way of stating Gauss' law?

[rudransh verma]

Gauss law relates the net flux phi of an electric field through a closed surface to the net charge q that is enclosed by that surface. It tells us that
Phi = q/permittivity
Can I say it like this : The gauss law states that the net flux of the surface depends upon the net charge enclosed by that surface and it does not depend upon the charge outside the surface.
The charge outside the surface would change the pattern of field but the net flux would not change because it is outside the surface.
So the Q charge outside would not enter the eqn of gauss law in any way.
It feels confusing to me.

 

[Gordianus]

I think it's a good way to state Gauss' law. As for the charge outside the closed surface, it's true. It doesn't contribute to the flux.

 

[rudransh verma]

I think it's a good way to state Gauss' law. As for the charge outside the closed surface, it's true. It doesn't contribute to the flux.

The enclosed charge has effect on net flux of the surface. That is gauss law. But the charge outside has no contribution to net flux. So we don’t write a law like phi=(q+Q)/permittivity.
I think we can insert a charge Q in the law but that will be of no use because the purpose of gauss law is to find Flux and E from net charge and vice versa. It is to find what is inside a box by looking at the box. Q is of no use here. It will only mess up the law. If Q is in the eqn we can never find total charge from the given flux. Also we cannot find field like this.

 

[rudransh verma]

The enclosed charge has effect on net flux of the surface. That is gauss law. But the charge outside has no contribution to net flux. So we don’t write a law like phi=(q+Q)/permittivity.
I think we can insert a charge Q in the law but that will be of no use because the purpose of gauss law is to find Flux and E from net charge and vice versa. It is to find what is inside a box by looking at the box. Q is of no use here. It will only mess up the law. If Q is in the eqn we can never find total charge from the given flux. Also we cannot find field like this.

@Delta2 can you verify this?

 

[Delta2]

@Delta2 can you verify this?

Well , tbh I don't know what would happen if Gauss's law was including the charge outside of the surface, I think one consequence would be that the Coulomb force wouldn't follow an inverse square law. And yes , as you say the whole point is to be able to calculate what's inside the box by looking at the box, we would lose this ability too.

 

[rudransh verma]

think one consequence would be that the Coulomb force wouldn't follow an inverse square law.

How?
e0E4pir^2=q+Q
E= 1/(4pie0)(q+Q)/r^2
Then?
I don’t follow!

 

[Delta2]

How?
e0E4pir^2=q+Q
E= 1/(4pie0)(q+Q)/r^2
Then?
I don’t follow!

Well, nvm, it seems I was wrong on that conclusion... I can't think of how you would define Q in that case, Q as just the charge outside, might as well been the charge of the whole universe.

 

[vanhees71]

The most simple and fundamental form of the electromagnetic laws are the local (differential) Maxwell equations. So it's always most simple to start from them. Gauss's Law for the electric fields read (using SI units)
$$\vec{\nabla} \cdot \vec{E}=\frac{1}{\epsilon_0} \rho.$$
Now use Gauss's integral theorem and integrate this equation over some volume $V$ with its boundary surface $\partial V$. This leads to
$$\int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{E}=\frac{1}{\epsilon_0} \int_{V} \mathrm{d}^3 r \rho=\frac{1}{\epsilon_0} Q_V.$$
This tells you that the flux of the electric field through a closed (!) surface is the charge $Q_V$ (times the ugly factor $1/\epsilon_0$ which is due to the choice of SI units) contained in the volume enclosed by this surface.

 

[rudransh verma]

Well, nvm, it seems I was wrong on that conclusion... I can't think of how you would define Q in that case, Q as just the charge outside, might as well been the charge of the whole universe.

This Q charge is enormous charge placed near the surface. Tell me does this field of charge Q only bend the fields of the charge inside the surface. I don’t think it amplifies/cut/intersect that field by interacting with it in any way. Because then the flux/density of the field of inside charge would change by an outside charge Q. It would then enter in gauss law.

 

[Delta2]

This Q charge is enormous charge placed near the surface. Tell me does this field of charge Q only bend the fields of the charge inside the surface. I don’t think it amplifies/cut/intersect that field by interacting with it in any way. Because then the flux/density of the field of inside charge would change by an outside charge Q. It would then enter in gauss law.

In physics we have a principle called the superposition principle. In the case of electric field, when we have two or more sources of electric field, then the field from each source, doesn't alter or interact the field from the other sources, but it just adds up, so we have a total field that is the sum of the fields of each source.

 

[rudransh verma]

In physics we have a principle called the superposition principle. In the case of electric field, when we have two or more sources of electric field, then the field from each source, doesn't alter or interact the field from the other sources, but it just adds up, so we have a total field that is the sum of the fields of each source.

I know. There is a resultant field with magnitude and direction. Book says the pattern of field changes. Doesn’t this effect the net flux of the surface?

 

[Delta2]

I know. There is a resultant field with magnitude and direction. Book says the pattern of field changes. Doesn’t this effect the net flux?

Yes, the pattern of the field in the surface changes, however this does not changes the net flux through the closed surface because whatever new flux enters our surface due to the outside charge Q, the same flux exits our surface through another segment of the surface, so the net flux =x+y-y=x+0=x where x the flux due to the inside charge and y the flux due to the outside charge.

 

[rudransh verma]

Yes, the pattern of the field in the surface changes, however this does not changes the net flux through the closed surface because whatever new flux enters our surface due to the outside charge Q, the same flux exits our surface through another segment of the surface, so the net flux =x+y-y=x+0=x where x the flux due to the inside charge and y the flux due to the outside charge.

But the flux x will change because the field of inside charge changes. Remember the resultant field. Now there is a new field with new magnitude and direction.

 

[vanhees71]

Yes, but the new field has no sources inside your volume. So its flux through the closed (!) surface must be 0.

 

[Delta2]

But the flux x will change because the field of inside charge changes. Remember the resultant field. Now there is a new field with new magnitude and direction.

No the flux x (which is the flux due to the field of the enclosed charge) does not change, because the field of the enclosed charge does not change. The resultant field at every point at our closed surface will be different, however the resultant net flux (if I can call it that way), will not change, because the new source is not enclosed by our closed surface.

Why this happens? I tried to argued in post #12 in terms of flux that enters and exits, but in fact there is no more why here, it is just what Gauss's law tell us, because Gauss's law tell us that when we calculate the net flux, it will be the same as long as the enclosed charge is the same, regardless of what happens to the resultant field.

 

[Delta2]

Please edit it. Your grammar seems wrong at some places.

Sorry I don't see anything wrong in my grammar at that post. I edited it though to put some paragraph formatting and some additional argument.

 

[rudransh verma]

No the flux x (which is the flux due to the field of the enclosed charge) does not change, because the field of the enclosed charge does not change. The resultant field at every point at our closed surface will be different, however the resultant net flux (if I can call it that way), will not change, because the new source is not enclosed by our closed surface.

Why this happens? I tried to argued in post #12 in terms of flux that enters and exits, but in fact there is no more why here, it is just what Gauss's law tell us, because Gauss's law tell us that when we calculate the net flux, it will be the same as long as the enclosed charge is the same, regardless of what happens to the resultant field.

You are saying the resultant field will change near the surface but the net flux will not because it only depends on charge inside the surface which is constant and outside charge contribute to zero flux. The flux x is the flux of the field that is coming from inside charge which will not change because charge doesn’t change. Inside charge is same. Only the resultant field is changing. The field from the inside charge doesn’t. So the flux x is not changing. It’s the same. So net phi= x+y-y=x.
We only talk of the flux of that field that is originating from a charge. There is still the same field from inside charge which can’t be altered. So it means there are three field vectors . One from outside charge, one from inside charge and the resultant.

 

[Delta2]

You are saying the resultant field will change near the surface but the net flux will not because it only depends on charge inside the surface which is constant and outside charge contribute to zero flux. The flux x is the flux of the field that is coming from inside charge which will not change because charge doesn’t change. Inside charge is same. Only the resultant field is changing. The field from the inside charge doesn’t. So the flux x is not changing. It’s the same. So net phi= x+y-y=x.
We only talk of the flux of that field that is originating from a charge. There is still the same field from inside charge which can’t be altered. So it means there are three field vectors . One from outside charge, one from inside charge and the resultant.

I agree to all of the above, seems to me you got it perfectly right!

 

[rudransh verma]

I agree to all of the above, seems to me you got it perfectly right!

It means the density of the field does not change if there is another field present. A test charge will move in the direction of resultant but the density of individual charge’s field stays same.

 

[Delta2]

It means the density of the field does not change if there is another field present. A test charge will move in the direction of resultant but the density of individual charge’s field stays same.

How do you define the term "density" of a field?

 

[rudransh verma]

I mean volume of field by density

 

[Delta2]

Ok I see, well, flux changes locally, because the resultant field changes locally, however the total net flux through a closed surface doesn't change if the additional source charges that cause the change of the resultant field are not within our closed surface.

 

[rudransh verma]

@Delta2 Can you summarise but don’t leave any detail. If one flux changes due to presence of another field then the net flux also change.

 

[Delta2]

@Delta2 Can you summarise but don’t leave any detail. If one flux changes due to presence of another field then the net flux also change.

OK I see now what you meant in post #19 and it is correct.

I understood something else: Suppose we have a charge q, and consider a gaussian spherical surface that surrounds q. Then the net flux is $\frac{q}{\epsilon_0}$. Now suppose we put another charge q' outside our spherical surface, outside the upper half of the surface. Then the total flux through the surface again remains the same $\frac{q}{\epsilon_0}$. However if I ask you for the flux of the upper half of our spherical surface, will it be the same or not?

 

[rudransh verma]

OK I see now what you meant in post #19 and it is correct.

I understood something else: Suppose we have a charge q, and consider a gaussian spherical surface that surrounds q. Then the net flux is $\frac{q}{\epsilon_0}$. Now suppose we put another charge q' outside our spherical surface, outside the upper half of the surface. Then the total flux through the surface again remains the same $\frac{q}{\epsilon_0}$. However if I ask you for the flux of the upper half of our spherical surface, will it be the same or not?

I think: the flux of upper half will change because of the changed field. So the net flux will change. I am not clear how two fields behave in each other’s vicinity

 

[Delta2]

I think: the flux of upper half will change because of the changed field. So the net flux will change.

The flux of the upper half will change indeed, but the net flux over all of our closed surface will not change because the enclosed charge doesn't change.

 

[rudransh verma]

@Delta2 I am not clear on how two fields interact in each other’s vicinity.
“Ok I see, well, flux changes locally, because the resultant field changes locally, however the total net flux through a closed surface doesn't change if the additional source charges that cause the change of the resultant field are not within our closed surface.”
why? If one flux changes then the net flux should also.

 

[Delta2]

I am not clear on how two fields interact in each other’s vicinity.

They just add up together to form a resultant field $\vec{E_{total}}=\vec{E_1}+\vec{E_2}$. The presence of $\vec{E_1}$ doesn't alter $\vec{E_2}$ neither vice versa.

 

[rudransh verma]

They just add up together to form a resultant field $\vec{E_{total}}=\vec{E_1}+\vec{E_2}$. The presence of $\vec{E_1}$ doesn't alter $\vec{E_2}$ neither vice versa.

What does this mean? That now only one field exist. Neither E1 or E2. Just Etotal. Or Are there three fields now.

 

[Delta2]

What does this mean? That now only one field exist. Neither E1 or E2. Just Etotal. Or Are there three fields now.

Well it depends how you see it. Mathematically there are three fields, but physically, yes there is only one field now, the total, in the sense that if we measure the E-field in a point in space we will find it to be $E_{total}$.

 

[rudransh verma]

Well it depends how you see it. Mathematically there are three fields, but physically, yes there is only one field now, the total, in the sense that if we measure the E-field in a point in space we will find it to be $E_{total}$.

So if we place a test charge it will move in the direction of new field.
So if now there is new field that means there is new flux in the upper half sphere and so new net flux.

“Ok I see, well, flux changes locally, because the resultant field changes locally, however the total net flux through a closed surface doesn't change if the additional source charges that cause the change of the resultant field are not within our closed surface.”
Why? If local flux changes then the net flux will also.

 

[Delta2]

So if we place a test charge it will move in the direction of new field.
So if now there is new field that means there is new flux in the upper half sphere and so new net flux.

Not necessarily new net flux. Net flux changes only if the charge that produces the new field, is inside our closed surface.

Why? If local flux changes then the net flux will also.

Not necessarily. Only if the local flux changes are due to new charges inside our closed surface.

In the case of my example at post #24, the change in flux of the upper half will be negated by a change in flux of the lower half and the total flux will remain the same.

 

[rudransh verma]

Only if the local flux changes are due to new charges inside our closed surface.

Again why it has to be inside charge?

 

[Delta2]

because that is what Gauss's law tell us: The flux through a closed surface changes only if the enclosed charge changes, regardless of what happens to the resultant field.
However the flux through an open surface can change if the resultant field is changing

 

[rudransh verma]

because that is what Gauss's law tell us: The flux through a closed surface changes only if the enclosed charge changes, regardless of what happens to the resultant field.
However the flux through an open surface can change if the resultant field is changing

How is that possible? If there is now a resultant field, if the field is changed now then the net flux must change. What kind of law is this?

 

[Delta2]

How is that possible? If there is now a resultant field, if the field is changed now then the net flux must change. What kind of law is this?

That's how Gauss's law goes whether you like it or not. Changes in the resultant field, result to changes in the net flux through a closed surface only if those changes in the resultant field are due to changes in the charges enclosed by the closed surface.

 

[vanhees71]

It's a mathematical theorem (see my posting above).

 

[Delta2]

It's a mathematical theorem (see my posting above).

Sorry I don't think this has anything to do with the divergence theorem. The divergence theorem is used to prove the equivalence of Gauss's law in integral and differential form.

 

[rudransh verma]

That's how Gauss's law goes whether you like it or not. Changes in the resultant field, result to changes in the net flux through a closed surface only if those changes in the resultant field are due to changes in the charges enclosed by the closed surface.

But you said the flux changes locally ie of one side. What about that?

 

[vanhees71]

Well yes, I tacitly assumed we take Maxwell's equations for granted. I hope we agree on this?

Then it's just Gauss's integral theorem that
$$\int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{E}=\frac{1}{\epsilon_0} \int_V \mathrm{d}^3 r \rho=\frac{1}{\epsilon_0} Q_V.$$

 

[vanhees71]

But you said the flux changes locally ie of one side. What about that?

What do you mean by "one side"? The flux in Gauss's law is always through a closed (!) surface!

 

[Delta2]

But you said the flux changes locally ie of one side. What about that?

Yes it can change in one side (for example upper half) but it won't change totally (through the whole closed surface), that is upper half+lower half, if the enclosed charge doesn't change. That is simply Gauss's law, changes in flux through a closed surface occur only when the enclosed charge changes. But changes in flux through an open surface can occur even if the enclosed charge doesn't change. Gauss's law is only for the net flux through closed surfaces.

 

[rudransh verma]

You are saying the resultant field will change near the surface but the net flux will not because it only depends on charge inside the surface which is constant and outside charge contribute to zero flux. The flux x is the flux of the field that is coming from inside charge which will not change because charge doesn’t change. Inside charge is same. Only the resultant field is changing. The field from the inside charge doesn’t. So the flux x is not changing. It’s the same. So net phi= x+y-y=x.
We only talk of the flux of that field that is originating from a charge. There is still the same field from inside charge which can’t be altered. So it means there are three field vectors . One from outside charge, one from inside charge and the resultant.

@Delta2 Let me make this more clear. You are saying the outside field from charge Q cannot alter the field inside the surface from enclosed charge. It can only add up with the charge q’s field only outside the surface. That’s where the flux changes locally. But the net flux remains same because the inside field is same as before being pierced by the charge Q. The field of charge Q enter and leave the surface. So phi= y-y+x=x. x is the flux of field inside the surface that never changes because enclosed charge never changes.

 

[rudransh verma]

@Delta2 I am unable to picture it properly.

 

[rudransh verma]

Yes, but the new field has no sources inside your volume. So its flux through the closed (!) surface must be 0.

Maybe now I got it. The new field has no lines of force. It’s just an effect of two fields due to charge q and Q, Q being outside the surface. There is no lines of force actually penetrating the surface from this new field. The only field of lines are from q andQ.There is no source charge inside the surface. Only source are q and Q combined. And so zero flux due to this field. And so the only fluxes are x, -y, +y which comes x as net flux. So Q charge is useless in gauss law.

 

[Delta2]

@Delta2 I am unable to picture it properly.

Sorry I just can't understand what exactly you don't understand. Gauss's law is for closed surfaces. For closed surfaces the total flux doesn't change (even if the resultant field changes due to an external charge) unless the enclosed charge changes, that's what Gauss's law tell us. For open surfaces, Gauss's law doesn't hold, so the flux over the open surface can change if the resultant field changes.

 

[rudransh verma]

Sorry I just can't understand what exactly you don't understand. Gauss's law is for closed surfaces. For closed surfaces the total flux doesn't change (even if the resultant field changes due to an external charge) unless the enclosed charge changes, that's what Gauss's law tell us. For open surfaces, Gauss's law doesn't hold, so the flux over the open surface can change if the resultant field changes.

Please read my post #45.

 

[Delta2]

Please read my post #45.

Well I read it and I don't understand why you want the resultant field to be some sort of inactive or invisible (without lines of force). Whether you speak of the resultant field, or the sum of the fields from q and Q, it is the same and one thing

 

[rudransh verma]

Well I read it and I don't understand why you want the resultant field to be some sort of inactive or invisible (without lines of force). Whether you speak of the resultant field, or the sum of the fields from q and Q, it is the same and one thing

Because only from a source can come out the field of lines. Like from q and Q. I didn’t say that it is inactive but it is the effect of two fields . Because of the presence of two fields this new field arises but not like there are some new lines of forces.
If we place one charge it will have lines of forces. A test charge will move in some direction. If we place another charge it will bend its line of forces and the direction of test charge also changes. So now there is new effect of the two fields.
Maybe there are new lines of forces but those lines don’t have a source inside the surface penetrating the surface.

 

[Delta2]

Well the resultant field has field lines that are the vector sum of the field lines of the two separate fields.

If we place another charge it will bend its line of forces

Yes the new bended line of forces are the force lines of the resultant field, where is the problem with this view?