2 Spheres of Charges and their Interaction

[Shreya]

Homework Statement

Please refer the image.

Relevant Equations

Nil

I can understand what happens with the conductor... (induction effects).
But how can induction happen in insulators ? Is it due to the the induced dipole moment?

 

[haruspex]

Certainly one would expect an induced dipole moment. What answer does that lead to? Do you know which answer is considered correct?

 

[Shreya]

moment. What answer does that lead to? Do you know which answer is considered correct?

The Option d is given as the right one.

 

[haruspex]

The Option d is given as the right one.

And is that what you would conclude from an induced dipole moment?

 

[Shreya]

And is that what you would conclude from an induced dipole moment

That is actually the part that I don't get. Please Help.

 

[haruspex]

That is actually the part that I don't get. Please Help.

Even in an insulator the charges can shift a little. E.g. at the level of an individual atom the electron cloud can be displaced slightly relative to the nucleus.
The charge movements are in the same directions as in a conductor, just much more restricted in magnitude.
So, for the purpose of the question, you can treat all objects as conductors.

Can you see how the induced charges change the force between the spheres for the two cases (like charges and unlike charges), or is that the part you don't get?

 

[Shreya]

Can you see how the induced charges change the force between the spheres for the two cases (like charges and unlike charges), or is that the part you don't get?

I think I get that.
In the case of like charges, say both positive (+ -)(+-) such charge configuration causes Fc>Fm
In the case of unlike charges, (+-)(+-) Fm>Fc

 

[haruspex]

I think I get that.
In the case of like charges, say both positive (+ -)(+-) such charge configuration causes Fc>Fm
In the case of unlike charges, (+-)(+-) Fm>Fc

Yes, but I can't see how those 'charge diagrams' illustrate it. I suspect the spacing got messed up by the website's formatting software. I presume you meant something like

.
In the case of like charges, say both positive (+...)(...+) such charge configuration causes Fc>Fm
In the case of unlike charges, (...-)(+...) Fm>Fc

 

[Shreya]

I suspect the spacing got messed up by the website's formatting software.

I think I messed it up.
Thanks a lot Haruspex for helping me out

 

[Delta2]

I am puzzled about something we take for granted but somehow it doesn't seem so obvious to me:
Let's assume that the two spheres do not interact via electrostatic induction, so no redistribution of charges on their surface. So there is uniform distribution of charge on their surface.

Why do we take for granted that the coulomb force in this case would be as if the whole charge was concentrated at the center of the spheres? We have to do integration to see it or is there some other argument?

 

[TSny]

I am puzzled about something we take for granted but somehow it doesn't seem so obvious to me:
Let's assume that the two spheres do not interact via electrostatic induction, so no redistribution of charges on their surface. So there is uniform distribution of charge on their surface.

Why do we take for granted that the coulomb force in this case would be as if the whole charge was concentrated at the center of the spheres? We have to do integration to see it or is there some other argument?

I don't think any integration is required if you already know (e.g., from Gauss' law) that a sphere with charge q uniformly spread on its surface produces an electric field outside the sphere that is identical to the field produced by a point charge q located at the center of the sphere.

See if you can argue without any mathematics that the force F in each figure below has the same magnitude.

Hopefully the meaning of F is clear in each diagram. For example, in the top diagram, F is the force on the right sphere produced by the left sphere. In the second diagram, F is the force on the right sphere if the left sphere is replaced by a point charge at the center of the left sphere. Etc.

 

[Shreya]

Let's assume that the two spheres do not interact via electrostatic induction, so no redistribution of charges on their surface. So there is uniform distribution of charge on their surface.

I think integration gives you the same result.
Using gauss law, by taking a spherical gaussian surface and computing the elctric field, will be much easier.
Another way to think is in terms of centre of charge (like centre of mass).
Every distribution of charges can be simplified to a point charge (position of the point charge will vary depending upon config). The sphere has a spherical symmetry (obviously!) So the uniformly distributed sphere can be considered to be a point charge located at the centre.
You may use symmetry arguments to state that the direction of the force will be along the line joining the centres of the sphere.

 

[Delta2]

Ok, @TSny I can see it now, I don't have to do integration but I 've to use Newton's 3rd law (if I am not mistaken) to conclude that the force between 2nd and 3rd picture is the same.

 

[TSny]

Ok, @TSny I can see it now, I don't have to do integration but I 've to use Newton's 3rd law (if I am not mistaken) to conclude that the force between 2nd and 3rd picture is the same.

Yes. Newton's third law holds for two point charges at rest. The superposition principle then implies that the third law holds for two arbitrary, static distributions of charge.