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Charge Conservation in a Pair of Concentric Spherical Objects


​​A small conducting sphere of radius $r_1 = 5.0$ cm is surrounded by a larger conducting spherical shell of radius $r_2 = 8.0$ cm, as shown in the figure below. The two spheres are concentric.

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A battery is connected between the inner sphere and the outer shell which moves (positive) charge from the inner sphere to the outer shell. After charging the spheres, the battery is disconnected. We measure the surface charge density of the outer shell and find that it is equal to $\sigma_{out} = +3.64$ $\mu\rm{C/cm^{2}}$.

What is the surface charge density $\sigma_{in}$ on the inner shell?


$\sigma_{in} = +3.64$ $\mu\rm{C/cm^{2}}$


$\sigma_{in} = -3.64$ $\mu\rm{C/cm^{2}}$


$\sigma_{in} = -9.32$ $\mu\rm{C/cm^{2}}$


$\sigma_{in} = -2.93$ mC/cm$^{2}$