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A spherical conductor A contains two spherical cavities as shown. The total charge on conductor itself is zero. However, there is a point charge q1 at the centre of one cavity and q2 at the centre of other. Another charge q3 is placed at a large distance r from the centre of the spherical conductor.

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Now my confusion is about the force exerted by q3 on the charges q1 and q2.

I think that q3 should exert exactly the same force on the other two charges as it would have exerted in the absence of the sphere and to nullify the aforesaid forces the induced charges on the sphere should exert an equal and opposite force on both the charges.

But its not what the case is, Why?

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2 Answers 2

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q1 and q2 induce charges of the same magnitude and of the opposite sign on the surface of the spherical cavities. This in turn means that on the outside surface of the sphere there are charges induced of the same magnitude and the same sign as charges q1 and q2. These charges are distributed on the outside surface of the spherical conductor. That is there is no way that just by looking at the surface charges can you deduce the location of the charge q1 and q2 or the location of the spherical cavities.

To find the force the relevant distance is that from the centre of the spherical conductor to charge q3. The charge on the spherical conductor is to be taken as q1 + q2.

Added later

This idea that the field from the sphere with charges inside it can be thought as the field from a single point charge at the centre of the conducting sphere can be explained as follows.
The surface of the sphere is an equipotential. E-field lines are always at right angles to an equipotential. So the E-field lines at the surface of the sphere are a normal to the sphere and so when produced backwards all appear to come from the centre of the conducting sphere.

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  • $\begingroup$ I am sorry but you did not answer my question: what about the force exerted by q3 on q2 and q1 ?. And secondly can you please explain the reason behind the uniform distribution of charges(q1+q2) on the surface of the sphere. $\endgroup$ Commented Jan 30, 2016 at 4:47
  • $\begingroup$ If $q1 = -q2$ then the best I can say is that there is no induced charges on the outside surface of the sphere and so no force between the sphere and charge q3. This is true no matter where inside the sphere q1 and q2 reside. In effect q3 cannot "see" charges q1 and q2 directly all it can "see" is the induced charge on the sphere. $\endgroup$
    – Farcher
    Commented Jan 30, 2016 at 8:28
  • $\begingroup$ My apologies. In this situation the induced charges will not be distributed uniformly over the sphere and I have now corrected my original answer. $\endgroup$
    – Farcher
    Commented Jan 30, 2016 at 8:31
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You're correct. And you're not. Depending on how you read the question. There's more than one way to interpret the phrase "the force exerted by q3 on q1."

You've interpreted it as just one in a bunch of terms in a net force equation, such that the net force on q1 is q1 times the sum of all the electric fields produced by all the charges in the problem. One of those terms is the usual Coulomb field between q1 and q3. Other terms happen to cancel that term out, and the net force on q1 is zero, but that' incidental; that q1q3 Coulomb term is still there regardless. Your interpretation makes sense and I think it's quite reasonable. Explicitly, I take it that you're interpreting "the force exerted by q3 on q1" to mean the force directly exerted, irrespective of all of the other terms cause by the surface charges. Under this (very reasonable) interpretation, your answer is quite correct.

But there's another way of interpreting the question. One might consider the phrase "the force exerted by q3 on q1" to mean "the net force exerted on q1 due to the presence of q3," which would of course be zero because (as you've surmised) the presence of q3 causes a rearrangement of surface charge such that the net effect on q1 is zero. And I suspect that this is how the majority of people would interpret it, to the extent that many people might not even realize that your understanding of the physics is fine but you're interpreting the question differently. It's a subconscious assumption. Under this interpretation (which I think is also a reasonable way to read the question), your answer would be wrong.

In short, it seems to me that you understand the physics just fine and the only problem is communication. The more I think about it, the more I think your interpretation is probably the better one even if it's not the first that comes to mind for many people. Still, I think both interpretations are reasonable and it may be of value for you to keep in mind both ways of describing the problem, and just be aware that there may be miscommunication because other people are interpreting the problem differently than you are.

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