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HW-10-solution

# HW-10-solution - HW-10 1(a Two metal objects are embedded...

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HW-10 1. (a) Two metal objects are embedded in weakly conducting material of conductivity . Show that the resistance between them is related to the capacitance of the arrangement by C R 0 (b) Suppose you connected a battery between the two conductors and charge them to a potential difference V0. If you then disconnect the battery, the charge will gradually leak off. Show that / 0 ) ( t e V t V , and find the time constant, , in terms of 0 and . Solution: (a) From the Ohm’s law, E J , we have the relation between the charge of the conductor, Q, and the current out of the conductor, I, by performing a surface integration over the conductor surface. 0 Q S d E S d E S d J I This is true for both conductors, i.e., all current flowing out of one conductor flows into another conductor. Then if V is the potential difference between the two conductors, we have C V Q Q V I V R 0 0 0 / / , where V Q C is the capacitance. (b) After disconnecting the battery, the change of the Q is the leak current. V CR V C I dt dQ C dt dV 0 1 The solution with the initial condition of V(t=0)=V 0 is an exponential function. t V t V 0 0 exp ) ( The time constant is 0 , which is roughly the time needed to leak out the charge. So the higher the conductivity is, the shorter time to leak off the charge.

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HW-10-solution - HW-10 1(a Two metal objects are embedded...

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