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Here is how I approached the question. Since the charge density on the two sphere are of opposite polarity therefore, electric field is zero in the mid way. This implies A is correct. Above and below the mid way line, magnitude of electric field is constant. This implies C is correct. As C is correct therefore, B is also correct.

But the answers according to the book are C and D.

I assumed the spheres are partially kept one over other and overlapping region is referred in the question.

Please help me to improve my concepts and solve the question.

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closed as off-topic by John Rennie, John Duffield, Kyle Kanos, peterh, AccidentalFourierTransform Feb 26 '17 at 20:41

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  • $\begingroup$ Just because the net charge in a region is zero does not mean the electric field in it is zero. Another example to consider is the electric field due a dipole where the charges are equal and opposite but still produce a field. Thus, an electric field is set up, conventionally directed from positive to negative and having same magnitude because of uniform charge density. Also, if you think the charges will neutralize, I don't think that is true since it is mentioned that the spheres are non-conducting. $\endgroup$ – LeroyJD Feb 25 '17 at 14:22
  • $\begingroup$ @LeroyJD I got why Electric Field is not zero. But can we say Electric Field decrease in magnitude on moving either side? $\endgroup$ – user411518 Feb 25 '17 at 14:28
  • $\begingroup$ the charge density remains uniform all throughout the sphere so why you think it will decrease? $\endgroup$ – LeroyJD Feb 25 '17 at 14:29
  • $\begingroup$ Could you update the title of your question to reflect that the spheres are overlapping? $\endgroup$ – ZeroTheHero Feb 25 '17 at 15:10
  • $\begingroup$ Please note that this is not a homework help site. Please see this Meta post on asking homework questions and this Meta post for "check my work" problems $\endgroup$ – Kyle Kanos Feb 26 '17 at 12:21
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This is a difficult problem. It hinges on the fact that, inside the individual spheres, the field is radial and linear with the distance, i.e. $\vec E(\vec r)\sim \vec r$. Thus, if $\vec R$ is the vector joining the two centres of the sphere, the electric field on the rightmost sphere will be $\sim \vec r_2=(\vec r_1-\vec R)$, where $\vec r_2$ is the location of a point as measure from the centre of the second sphere.

Thus, in your question, the field of the first sphere at $\vec r_1$ minus the field of the second sphere at $\vec r_1-\vec R$ will give you a constant field along $\vec R$.

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I don't think the electric field is zero in the middle. My argument for this is because the smaller sphere has a smaller radius it therefore has a larger charge density per unit volume than the greater sphere, if they have the same total charge density. From the figure this would mean that there is a more negative charge than positive charge in the overlapping region. And since we now have concluded that the electric field is non zero in the overlapping region the electrostatic potential can't be constant.

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