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By definition the relation between electric field and electric potential at any given point in space is
Inside a conducting shell the electric field is zero (otherwise the field will move the charges inside the sphere). Because of this and of the above relation between field and potential we can deduct that inside a conducting shell the potential is constant.
Because the field is zero inside the shell its flux through the shell is zero.
`E_i =0 rArr Phi =0`
If we take the surface for which the flux is computed having the form of the shell walls we deduct
`Phi = Q_i/epsilon_0 =0 rArr Q_i =0`
Thus the sum of the charge inside the shell and on the inner walls of shell need to be zero.
`Q_i = Q_"InnerSurface" -3C =0 rArr Q_"InnerSurface" =+3C`
Now we know the total charge on the shell is 10 C.
`Q_"OuterSurface" +Q_"InnerSurface" =10 C rArr Q_"OuterSurface" =7 C`
The net charge on the outer surface of the conducting shell is +7 C.
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