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If we enclose a negatively charged sphere inside a positively charged sphere, do we get the electric field only due to the outer sphere?

If we do, how? Also, where did the negative charge went ?

enter image description here

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No. The field outside the outer sphere is a superposition of the electric field produced by the positive charge and the field produced by the negative charge. Since the field produced by a uniformly charged spherical shell is the same as that of a point charge at the center (for points outside the sphere), the field outside the outer sphere will be the same as though the entire charge of both spheres was turned into a point charge at their common center. In particular, if the two spheres have equal and opposite charges, the field outside the outer sphere will be zero.

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  • $\begingroup$ Indeed, this is a classisical example of the situation for a neutral noble gas atom (with charges reversed) $\endgroup$ – Lewis Miller Jun 21 '17 at 14:16
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If you know Gauss Law then this problem is a piece of cake. From Gauss Law we know that the net electric flux comming out through an enclosed volume is equal to some constant times the net charge within the volume. Hence electric field would be due to the contribution of both the charges in inner and outer sphere outside the region of the bigger sphere.

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In terms of a picture with your 15 negative charges on the inner sphere and your 20 positive charges on the outer sphere, the charges rearrange themselves as follows.

enter image description here

You will have 15 negative charges on the outer surface of the inner sphere and 15 positive charges on the inner of the outer sphere.
The remaining five positive charges will reside on the outer surface of the outer sphere.
There will be no charges resident inside the two conducting spheres.

The electric field will be zero inside the inner sphere , an electric due to the 15 negative charges (and 15 positive charges) between the two spheres and an electric field due to 5 ($= 20-15$) positive charges outside the outer sphere.

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