Question

A conductor is made of an isotropic material (resistivity $\rho$) has rectangular cross section. The horizontal dimensions of the conductor decreases linearly from 2x at one end to x at the other end and vertical dimensions increase from y to 2y as shown in the figure. The length of conductor along the axis is equal to l and a battery is connected across the conductor, then

• A. resistance of conductor is equal to $\frac{4\rho l}{9xy}$
• B. resistance of conductor is equal to $\frac{9\rho l}{4xy}$
• C. drift velocity of conduction electrons is maximum at the middle section
• D. drift velocity of conduction electrons is minimum at the middle section

Answer 1

The correct option is D drift velocity of conduction electrons is minimum at the middle section

The drift velocity, U=I/NAQ . The Current I , Charge Density N , and the Charge magnitude Q is constant. The only variable is the area. Let the length along the axis be z with z=0 at the first face. The area A could be then written as (y+yz)(2x-xz) = xy(1+z)(2-z). Here, x and y are constants and the variable is z . Differentiating (1+z)(2-z) w.r.t z and equating to 0 we get the maxima at z=0.5 . Thus, the area is maximum at z=0.5 and the drift velocity U is therefore minimum at z=0.5 .