On charge image method over a conducting sphere and its Char

[Espoice]

Hi colleagues,

I need your help to figure out some subtle theory problmes I have about this topic.

I was thinking about a system composed by a conductive sphere (not grounded) under the effect of a pont charge placed near the sphere. As I know we need two image charges to satisfy the boundary conditions of the system, precisely a constant potential over the sphere surface and its value going to zero when distance from the system approaches infinity. Actually we need two opposite image charges but equals in modulus, one of these placed in the center of the sphere. This is convincing because the actual flux round the sphere is zero as it should be for a conductive finte object.

But when it comes the sphere to be grounded, the problem is solved placing just one image charge inside the sphere to satisfy the condition of having potential equals zero throughout the sphere surface. This setting however does not lead to nullify the flux valuated on the sphere surface, as I think it should be equal to zero. How is this explained?

Can anyone find flaws in my reasoning? Is there something I am missing?

Really appreciate any help! Have a good day!

 

[Orodruin]

A grounded sphere does not have fixed charge. An ungrounded one does.

Edit: changed choice of words.

 

[mfb]

If the sphere is grounded it can (and will) acquire a non-zero total charge.

 

[Espoice]

Thanks for the quick reply!
So actually when the sphere is grounded it happens that all the opposite charges inside the sphere, with respect to the one which generates the field, goes away through the ground so the sphere will remain actually charged?

 

[Orodruin]

Yes.

 

[Espoice]

A grounded sphere does not have fixed charge. An ungrounded one does.

Edit: changed choice of words.

So we state there is some kind of non zero flux over the sphere when the sphere is grounded. Does this mean we have an electric field inside the sphere? Can you please clarify this passage?
Thanks again!

 

[Orodruin]

So we state there is some kind of non zero flux over the sphere when the sphere is grounded. Does this mean we have an electric field inside the sphere? Can you please clarify this passage?
Thanks again!

No, the situation is static. There will be no actual non-zero field inside the sphere (it is a conductor!). The method of images is solving for the field outside the sphere. The mirror images has nothing to do with the actual charge distribution inside the sphere.

The point is that in the ungrounded sphere you know what the overall charge should be (by Gauss law the total of the mirror charges must be equal to the actual total surface charge on the sphere). In this case zero. In the grounded case you do not know the total charge in the static situation since charge can flow to/from the ground.