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The cross section of a cylindrical conductor is A. The resistivity of the material of the cylinder depends only on distance r from the axis of the conductor as rho=k/r^2 where k is a constant. Find the resistance per unit length of such a conductor. |
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Answer» SOLUTION :Let US CONSIDER a THIN cylindrical pipe of unit length having radius r and thickness dr. The conductance of thinpipe is , `dY=(1dA)/(rho 1)`.......(1) when dA= cross area of cylindrical pipe.  Here, `dA=2pir d r` `therefore` From equation (1), `dY=1/rho.2pirdr` `becauserho=k/r^2so,dY=1/k.2pir^3dr` Therefore, electrical conductance of the whole conductor becomes, `Y=(2pi)/kint_0^ar^3dr` [a= radius of the whole conductor] `=(2pi)/ktimes[r^4/4]_0^a=(pia^4)/(2k)=A^2/(2pik)[becauseA=pia^2]` So, resistance`=1/Y=(2pik)/A^2` |
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