A current carrying cylindrical conductor PQ is lying along the x-axis with its end P at x = 0. The cylinder has a uniform cross-section and it is made of an alloy whose resistivity varies with x as , 

                        $\mathrm{\rho }=\mathrm{ax}+\mathrm{b}$

where a and b are positive constants. Which of the following graphs represents the variation of potential difference ( V) with x? 

By Ohm's law, the current density J and electric field E are related as:

  $$\begin{gathered} J=\sigma E=\frac{E}{\rho }; \left(\sigma =conductivity\right) \\ \therefore E=\rho J=\left(ax+b\right)J \end{gathered}$$

Since the cross-sectional area is constant, J is constant. 

Now                   $E=-\frac{dV}{dx}$

$\Rightarrow dV=-Edx$

Integrating:

       ${\int }_0^{V}dV=-{\int }_0^x\left(ax+b\right)Jdx$

$$\begin{gathered} \Rightarrow \mathrm{V}-0=-\mathrm{J}\left(\frac{{\mathrm{ax}}^2}{2}+\mathrm{bx}\right) \\ \mathrm{or}, \mathrm{V}=-\mathrm{J}\left(\frac{{\mathrm{ax}}^2}{2}+\mathrm{bx}\right) .................. \left(1\right) \end{gathered}$$

Eq. (1) is the equation of a symmetric parabola (as in the case of the trajectory of a projectile)