Electric Field Inside Conductor: Showing Zero is Only Value Possible

[HungryChemist]

Field inside conductor in electrostatic is zero. So they say...cuz the free-particle will move due to eny Electricfield present until they all stablized to the surface of the conductor to cancel the Electricfield present inside, thereby making the net field inside zero.

After understading the argument such as above, I set my work to show the electric field inside conductor is zero(Given total charge inside of the conductor but not the surface is zero) rather mathematically and got nowhere. Any help or advise will be appreciated.

If I already know total charge inside of the conductor is zero, from Gauss's law it follows that Electric field inside of conductor must be constant value. So it means that it must be either zero or non-zero constant value. Now, how can I show that zero is the only value possible?

 

[Tide]

If the electric field inside the conductor were a nonzero constant then electrical current (from both positive and negative charges) would flow to the surface and charge would build up indefinitely.

 

[heman]

Field inside conductor in electrostatic is zero. So they say...cuz the free-particle will move due to eny Electricfield present until they all stablized to the surface of the conductor to cancel the Electricfield present inside, thereby making the net field inside zero.

Simpler way to say this will be,
If there were any field the free charges would move and there wouldn't be Electrostatics anymore.
The induced charges play the role here,the field due to them tends to cancel off the original field.

After understading the argument such as above, I set my work to show the electric field inside conductor is zero(Given total charge inside of the conductor but not the surface is zero) rather mathematically and got nowhere. Any help or advise will be appreciated.

Well the surface is the only place for charge which allows them to stay.
When you put conductor in an external electric field,this drives the positive and negative charges apart like positive to right and negative to left if E is to the right.Hence an electric field develops which is equal to the external electric field.That Leads to ZERO.
There is no such great mathematics involved here,as you can see by formula of induced charges it is exactly equal and opposite to charge producing external electric field.This looks simple i guess.

If I already know total charge inside of the conductor is zero, from Gauss's law it follows that Electric field inside of conductor must be constant value. So it means that it must be either zero or non-zero constant value. Now, how can I show that zero is the only value possible?

If total charge is zero Electric field is bound to zero.Where do you get constant in Gauss Law in such a case?
$$\oint{\vec{E} \cdot d\vec{a}} = \frac{Q_{enc}}{\epsilon_0}$$
There is no constant of integration here!

 

[HungryChemist]

If total charge is zero Electric field is bound to zero.Where do you get constant in Gauss Law in such a case?
$$\oint{\vec{E} \cdot d\vec{a}} = 0$$
There is no constant of integration here!

$$\oint{\vec{E}\cdot d\vec{a}} = 0$$

doesn't say that $$\vec{E} = 0$$. It could be that E is zero or E dot dA is zero. For example, if the Electric field is constant over the region of Gaussian surface then we still have $$\oint\vec{E}\cdot d\vec{a}} = 0}$$.



I was wondering if one can show explicitly(from math point of view) that $$\vec{E} = 0$$ is necessary result.

I didn't say the total charge is zero, I said the charge inside of the gaussian surface which in this case the entire volume of the conductor except at the surface of conductor itself the charge is zero.

 

[Doc Al]

$$\oint{\vec{E}\cdot d\vec{a}} = 0$$

doesn't say that $$\vec{E} = 0$$.

The only way that that integral equation can be satisfied for any conceivable closed surface within the conductor is if E = 0 everywhere.

 

[HungryChemist]

The only way that that integral equation can be satisfied for any conceivable closed surface within the conductor is if E = 0 everywhere.

Can you explain further? If the electric field is constant in magnitude and direction, the above closed surface integral will still give zero. That is why I mentioned that E could be either zero or constant value. What makes you say that E could be zero only?

 

[Claude Bile]

If there is an E field present, currents will flow within the conductor until the E field is zero.

There is no mathematical argument as to why E must be zero, only physical ones. For example, in the case of an insulator, the situation is different, despite the fact the same equation is involved.

Claude.

 

[heman]

Can you explain further? If the electric field is constant in magnitude and direction, the above closed surface integral will still give zero. That is why I mentioned that E could be either zero or constant value. What makes you say that E could be zero only?

I wonder what makes you think that E could be constant value.If you would have been saying E is perpendicular to area vector's direction then that would have made sense.

 

[Doc Al]

Can you explain further? If the electric field is constant in magnitude and direction, the above closed surface integral will still give zero. That is why I mentioned that E could be either zero or constant value. What makes you say that E could be zero only?

You are, of course, correct. My statement in post #5 is nonsense. What I meant to say was that that integral equation can be satisfied for any conceivable closed surface within the conductor only if the charge = 0 everywhere within the conductor. But, of course, that was not your question! (So I guess I made two errors in that post. )

Claude Bile is correct; the argument for E = 0 within a conductor is a physical one.

 

[HungryChemist]

You are, of course, correct. My statement in post #5 is nonsense. What I meant to say was that that integral equation can be satisfied for any conceivable closed surface within the conductor only if the charge = 0 everywhere within the conductor. But, of course, that was not your question! (So I guess I made two errors in that post. )

Claude Bile is correct; the argument for E = 0 within a conductor is a physical one.

After all, I don't speak English well. Thanks for your reply!

I knew the physical explanation since that was only explanation I could find from Grifitth and Feynmman but I thought that they were skipping the mathematical arguement.

But still, it doesn't quiet make sense to me. So if I put a single particle with some charge inside conductor, that particle will not accerlate because there are no electric fields present inside of conductor? But for the other particles, now there are electric field inside conductor due to the particle that I just put in?

 

[HungryChemist]

I wonder what makes you think that E could be constant value.If you would have been saying E is perpendicular to area vector's direction then that would have made sense.

Divergent of E is equal the total Charge inside. (Gauss's law)

If the total Charge inside is zero(since that is what I am assuming) the Divergent of E must also be equal to zero. Which tells me that E must be constant (whether zero or not)

I am sorry for not being able to write clearly, I was simply trying to derive the result that inside of conductor Electric field must be zero given that charge inside is zero. Most of the texts that I've been reading along with do this conversly. They say E field inside of conductor is zero then derive the result that the charge inside of conductor must also be zero.

Like above people point out, maybe there can't be given a mathematical argument to show Electric field must be zero in that closed surface integral being equal to zero. Only the physical one. But there might be just one, one can not know such thing for sure until he shows that it is impossible to show. Right?

 

[Doc Al]

So if I put a single particle with some charge inside conductor, that particle will not accerlate because there are no electric fields present inside of conductor? But for the other particles, now there are electric field inside conductor due to the particle that I just put in?

I don't know what you mean by "put a particle with some charge inside a conductor". In a conductor, the charge carriers are free to distribute themselves as needed.

 

[HungryChemist]

I don't know what you mean by "put a particle with some charge inside a conductor". In a conductor, the charge carriers are free to distribute themselves as needed.

Say I have a conductor of spherical shell. We claim that inside of such conductor the electric field is zero. If I can put an electron inside of such conductor, surely the electron will not distribute itself because there isn't any Electric field to move such electron since it is a conductor. So, I imagined that the electron will just sits inside of conductor where I put it. Then this conductor is not a conductor anymore becouse now there's electric field insdie due to the electron I just put it in. Is this a total nonsese?

 

[Doc Al]

Say I have a conductor of spherical shell. We claim that inside of such conductor the electric field is zero. If I can put an electron inside of such conductor, surely the electron will not distribute itself because there isn't any Electric field to move such electron since it is a conductor. So, I imagined that the electron will just sits inside of conductor where I put it. Then this conductor is not a conductor anymore becouse now there's electric field insdie due to the electron I just put it in. Is this a total nonsese?

Again, I don't know what you mean by "putting an electron in a conductor". (How would you do that?) Are you talking about a hollow conducting shell? And putting the electron within the hollow?

If you could magically fix an extra electron inside a conductor, then the other charge would just redistribute themselves to cancel the field.

 

[Meir Achuz]

If you put an electron into a conductor, its electric field would polarize the charge distribution on the surface of the conductor. This redistributed charge would attract the electron to the surface of the conductor in about 10^-19 seconds, leaving no field and no electron inbside the conductor.

 

[Creator]

If you put an electron into a conductor, its electric field would polarize the charge distribution on the surface of the conductor. This redistributed charge would attract the electron to the surface of the conductor in about 10^-19 seconds, ...

Really? Let's see... for an electron placed 0.1 meter from the surface, it would have a velocity of :
V = d/t = 0.1m/10^-19 sec.

V = 10^18 meters/sec.

Hmm; not bad for those of us trying to create superluminal electrons.

 

[Doc Al]

Don't think of that single electron as working its way to the surface at superluminal speed. Instead, think of the conductor being filled with a "sea of electrons" which shift ever so slightly, which happens very quickly. The net result is that the surface of the conductor has gained one electron's worth of charge (but the electron placed in the middle doesn't have to travel to the surface).

 

[Claude Bile]

Divergent of E is equal the total Charge inside. (Gauss's law)

If the total Charge inside is zero(since that is what I am assuming) the Divergent of E must also be equal to zero. Which tells me that E must be constant (whether zero or not)

I am sorry for not being able to write clearly, I was simply trying to derive the result that inside of conductor Electric field must be zero given that charge inside is zero. Most of the texts that I've been reading along with do this conversly. They say E field inside of conductor is zero then derive the result that the charge inside of conductor must also be zero.

Like above people point out, maybe there can't be given a mathematical argument to show Electric field must be zero in that closed surface integral being equal to zero. Only the physical one. But there might be just one, one can not know such thing for sure until he shows that it is impossible to show. Right?

Gauss Law applies to all materials, conducting or otherwise. How can you mathematically show that the electric field inside a conductor is zero, when, mathematically the field inside an insulator is not necessarily zero? You have to take into account that you are dealing with a conductor somewhere in your derivation.

Claude.

 

[HungryChemist]

Gauss Law applies to all materials, conducting or otherwise. How can you mathematically show that the electric field inside a conductor is zero, when, mathematically the field inside an insulator is not necessarily zero? You have to take into account that you are dealing with a conductor somewhere in your derivation.

Claude.

I am not sure if I understand what you're talking about. My original or intended task was to show (mathematically, starting from Gauss's law) the inside of conductor the electric field must be zero given that there are no charge inside. Of course, I don't know how to do it and that's why I was seeking advice. To be honset, I didn't get anywhere so there isn't really any derivation I've worked out. All I found was rather ovious result which says that inside of such conductor the Electric field must be constant. How should I take into account that I am dealing with a conductor so that I can finally show the field inside of such conductor must be zero? At this point I really gave up

 

[Claude Bile]

Okay, I will try to explain my previous post in simpler terms.

Say you wish to apply Gauss law for the case where the enclosed charge is zero, for the case of both a conductor and insulator. If you could show mathematically that the only solution in this case was E = 0, then this solution would apply to BOTH the Conductor and the Insulator.

However, we know the solution E = 0 does not, in general apply for an insulator. In an insulator, the material polarises which requires things like displacement fields to be considered.

Hence upon arriving at the solution E = 0 for a conductor, you must, at some point make some assertion to separate the conductor case from the insultor case. In the case of the conductor, the assertion is made that the conductor contains free charges which are free to move and thus cancel out any electric field.

I hope that makes a bit more sense.

Claude.

 

[Meir Achuz]

Field inside conductor in electrostatic is zero. So they say...cuz the free-particle will move due to eny Electricfield present until they all stablized to the surface of the conductor to cancel the Electricfield present inside, thereby making the net field inside zero.

After understading the argument such as above, I set my work to show the electric field inside conductor is zero(Given total charge inside of the conductor but not the surface is zero) rather mathematically and got nowhere. Any help or advise will be appreciated.

If I already know total charge inside of the conductor is zero, from Gauss's law it follows that Electric field inside of conductor must be constant value. So it means that it must be either zero or non-zero constant value. Now, how can I show that zero is the only value possible?

The first paragraph is correct (except for spelling). It gives the reason that there can be no static electric field inside a conductor.

I don't understand the motivation of the second paragraph. If something is shown so simply, why look for a more complicated reason (using circular logic)?

The third paragraph is wrong. Gauss's law relates an integral to an integral. This means that it cannot be used to say anything about E at each point inside a conductor.