A cylindrical solid of length L and radius a is having varying resistivity given by `rho = rho_(0) x`, where` rho_(0)` is a positive constant and x is measured from left end of solid. The cell shown in the figure is having emf V and negligible internal resistance. The electric field as a function of x is best described by

Options

A. `(2 V)/(L^(2) ) x`
B. `(2 V)/( rho_(0) L^(2) ) x`
C. `(V )/( L^(2) ) x`
D. None of these

Answer

A

Explanation

Consider an element part of solid at a distance x from left end of width dx.

Resistance of this elemental part is,
`d R= ( rho dx)/ (pi a^(2)) = ( rho_(0) x dx)/( pi a^(2))`
`R= int d R = int_(0)^(L) (rho_(0) x dx)/( pi a^(2)) (rho_(0) L^(2) )/ (2 pi a^(2))`
Current through cylinder is, `I= (V)/( R) = ( V xx 2 pi a^(2))/( rho_(0) L^(2))`
Potential drop across element is, `d V = (1 d R)= ( 2 V)/( L^(2) ) xdx`
`E (x)= (dV)( dx) = (2 V)/( L^(2)) x`