Confusion on the distribution of charge

[Gourab_chill]

Homework Statement

I'm interested in knowing the distribution of the charges of the figure given in the attachments.

Relevant Equations

columb's formula of electrostatics

The charges are q1,q2 & q. P,Q,O1,O2 refer to positions only. This is a conducting sphere with cavities containing charges.

I'm interested in knowing how the charge should be distributed in the sphere. I know the charges induced on the charges of the sphere should be equal and opposite to the charge contained by them(q1 & q2). But there is also a charge q outside, so there should be a equal and opposite charge formed on the surface of the sphere, right?

But if I do that then the total induced charges are going to be unbalanced violating the law of conservation of charge. Where am I wrong?

 

[haruspex]

Consider a simple charged conducting block, no cavities. Where will the charge reside?
Now create a cavity, but no charge inside it. Is there any reason for some charge to migrate to the surface of the cavity?

 

[Gourab_chill]

Consider a simple charged conducting block, no cavities. Where will the charge reside?
Now create a cavity, but no charge inside it. Is there any reason for some charge to migrate to the surface of the cavity?

No. Until there is some charge in the cavity, I think.

Another clarification I wanted to make on the second para, first line(of the main question body):

I know the charges induced on the charges of the sphere should be equal and opposite to the charge contained by them(q1 & q2).

Actually this line has typos in it and I regret for not checking will I was typing it. I wanted to type:

"I know the charges induced on the surfaces of the cavities should be equal and opposite to the charges contained by them(q1 & q2)."

Since in the original question there is no charge on the original block; and there is some charges in the cavities and also in the exterior, how should the charges on the block distribute? I do have a idea that the induced charges on the surface of the cavities will try to negate the electric field due to the cavity charges, and so will the induced charges on the outer surface of the block due to the exterior charge q?

Sorry for the inconvenience caused due to typing mistakes.

 

[haruspex]

Since in the original question there is no charge on the original block; and there is some charges in the cavities and also in the exterior, how should the charges on the block distribute? I do have a idea that the induced charges on the surface of the cavities will try to negate the electric field due to the cavity charges, and so will the induced charges on the outer surface of the block due to the exterior charge q?

Charges having migrated to the cavity boundaries to neutralise the charges therein, the remaining block has a changed total charge, but feels no field from the internal charges. So the change to the charge distribution on the outer faces of the block will be multiplication by a constant.

 

[Gourab_chill]

So the change to the charge distribution on the outer faces of the block will be multiplication by a constant.

I couldn't get what you meant by 'multiplication by constant', is it the net charge (q1+q2) you mean?

 

[haruspex]

I couldn't get what you meant by 'multiplication by constant', is it the net charge (q1+q2) you mean?

If a conductor has charge Q, that distributes in some way around its boundary.
If we add another Q' of charge, a patch of the boundary that had charge q will now have charge $q\frac {Q+Q'}Q$. In other words, the extra charge distributes in the same pattern as the original charge.

 

[Gourab_chill]

If a conductor has charge Q, that distributes in some way around its boundary.
If we add another Q' of charge, a patch of the boundary that had charge q will now have charge $q\frac {Q+Q'}Q$. In other words, the extra charge distributes in the same pattern as the original charge.

So, can I conclude from this is that the charge will distribute in such a way that there will opposite charge along the patch facing the charge q such that the electric field by it(charge q) will not be expressed inside the conductor/block and the charge on the the other patch of the block will recreate the electric field by q as it should have been at that point onwards?

Am I wrong or right?

 

[haruspex]

So, can I conclude from this is that the charge will distribute in such a way that there will opposite charge along the patch facing the charge q such that the electric field by it(charge q) will not be expressed inside the conductor/block and the charge on the the other patch of the block will recreate the electric field by q as it should have been at that point onwards?

Am I wrong or right?

Is this in the context of the original question, i.e. q is an external point charge?
If so, yes, the charges on the surface of the block will arrange to neutralise the field from q within the block. But I did not understand the last part. By "other patch" do you mean the rest of the surface? It won’t recreate the field from q, as though the block were not there.

For some insight into how a conductor distorts a surrounding field see fig 6c at https://courses.lumenlearning.com/p...rs-and-electric-fields-in-static-equilibrium/