Two conducting layers of length `L` each specific resistivites `rho_(1)` and `rho_(2)` and cross-sectional areas `A_(1)` and `A_(2)` respectively are connected to a voltage source `V` such that current flow lines do not cross from one layer to the other.
(i). Find the total current in the system and describe how it is divided between the 2 layers
(ii). Find the total current when a third layer of cross-sectional area `A_(1)+A_(2)`, length `L` and specific resistivity `rho_(3)` is attached at one of the ends of the given system.

Answer

(i). `((v)/(R_(1)R_(2))(R_(1)+R_(2)))xxR_(1)`
`((v)/(R_(1)R_(2))(R_(1)+r_(2)))R_(2)`
(ii). `(v)/((R_(1)R_(2))/(R_(1)+R_(2))+R_(3)` where `R_(1)=(rho_(1)l)/(A_(1))`

Explanation

(i) resistance of the 2 layers are:
`R_(1)=(L)/(A_(1))rho_(1),R_(2)=(L)/(A_(2))rho_(2)`
Hence, total resistance across the voltage source is: