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The space betweenthe platesof a parallel -platecapacitoris filled upwith inhomogneous poorlyconductingmedium whoseresistivity varieslinearlyin the directionperpendicularto theplates. The ratio of the maximumvalueof resistivity to the minimumone is equalto eta The gapwidthequals d. Find thevolume densityof the chargein the gap if a voltage V is appliedto the capacitor. epsilon is assumedto be 1 everywhere. |
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Answer» Solution :As in the PREVIOUS problem `E_(X) = C rho (x) = C (rho_(0) - rho_(1) x)` where `rho_(0) + rho_(1) d eta rho_(0)` or,`rho_(1) = ((eta - 1) rho_(0))/(d)` By intergation `V = int_(0)^(d) C rho (x)dx = Crho_(0) d (1 + (eta - 1)/(2)) = (1)/(2) C rho_(0) d (eta - 1)` Thus `C = (2V)/(rho_(0) d (eta + 1))` Thus volume density of charge present in the medium `= (DQ)/(sdx) = epsilon_(0) dE (x)//dx` `= (2 epsilon_(0) V)/(rho_(0) d (eta + 1)) xx ((eta - 1) rho_(0))/(d) = (2epsilon_(0) V(eta - 1))/((eta + 1) d^(2))` |
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