Posted: Wed Dec 28, 2011 3:40 pm
kiteman is on the right track. i think the confusion here is we are talking electrodynamics where you are talking electrostatics. we are both correct.
electrostatically, the charges in the conductor will move so as to neutralize any voltage gradients, thus moving so as to make themselves invisible to anything on the inside.
however, we are keeping the outer sphere at ground, and the inner sphere at a constant positive voltage relative to ground. as just mentioned, the charges will move so as to neutralize the voltage gradient. this means electrons will move from the outer sphere to the inner sphere. thus, we will not have this charge differential for very long. to get it back we have to move electrons from the inner sphere back to the outer sphere. when we say we are holding the inner and outer spheres at constant charge, that implies that we are doing precisely this.
so you see, while gauss' law (of course) accurately tells us what the system will be like when it reaches equillibrium, we are actively keeping the system _out_ of equilibrium by applying a current between the inner and outer spheres. this current constantly recharges the voltage gradient that the electrons are constantly neutralizing. if we turn the power off -- if we either unground the outer sphere or stop pumping electrons out of the inner sphere -- the electrons will quickly succeed in neutralizing the voltage gradient and we'd come to the situation described by you; by gauss's law. which means we would no longer have a charge differential between the two spheres (well, at least not one that can't be maintained electrostaticaly for very long). and the situation ipso facto would no longer be that which we seek to analyze.
electrostatically, the charges in the conductor will move so as to neutralize any voltage gradients, thus moving so as to make themselves invisible to anything on the inside.
however, we are keeping the outer sphere at ground, and the inner sphere at a constant positive voltage relative to ground. as just mentioned, the charges will move so as to neutralize the voltage gradient. this means electrons will move from the outer sphere to the inner sphere. thus, we will not have this charge differential for very long. to get it back we have to move electrons from the inner sphere back to the outer sphere. when we say we are holding the inner and outer spheres at constant charge, that implies that we are doing precisely this.
so you see, while gauss' law (of course) accurately tells us what the system will be like when it reaches equillibrium, we are actively keeping the system _out_ of equilibrium by applying a current between the inner and outer spheres. this current constantly recharges the voltage gradient that the electrons are constantly neutralizing. if we turn the power off -- if we either unground the outer sphere or stop pumping electrons out of the inner sphere -- the electrons will quickly succeed in neutralizing the voltage gradient and we'd come to the situation described by you; by gauss's law. which means we would no longer have a charge differential between the two spheres (well, at least not one that can't be maintained electrostaticaly for very long). and the situation ipso facto would no longer be that which we seek to analyze.