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An inhomegenous poorly conducting medium fills up the spacebetween plates 1 and 2 of parallel-platecapacitor. Itspermittivityand resistivityvary from values epsilon_(1), rho_(1) at the plate 1 to vlaues epsilon_(2), rho_(2) at plate 2. A dcvoltageis appliedto the capacitorthrouggh which a steadycurrent I flowsfrom plate1 to plate 2. Find the totalextraneous charge in the given medium. |
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Answer» Solution :By current conservation `(E (X))/(rho (x)) = (E (x) + d E(x))/(rho (x) + d rho (x)) = (d E (x))/(d rho(x))` This has the solution, `E_(x) = C rho (x) = (I rho (x))/(A)` Hence charge induced in the SLICE per unit area `d sigma = epsilon_(0) (I)/(A) [{epsilon (x) + d epsilon (x)} {rho (x) + d rho (x)} - epsilon(x)rho (x)] = epsilon_(0) (I)/(A) d [epsilon (x) rho (x)]` Thus, `dQ = epsilon_(0) Id [epsilon (x) rho (x)]` Hence total charge induced is by intergation, `Q = epsilon_(0) I (epsilon_(2) rho_(2) - epsilon_(1) rho_(1))` 
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