Resistivity And Wire Geometry

Resistance depends on geometry through the relation $R=\rho \frac{l}{A}$, where $\rho$ is resistivity, $l$ is length, and $A$ is cross-sectional area, as established in NCERT Class 12, Chapter 3 (Current Electricity). When a wire is stretched at constant volume, $V=Al$ stays fixed, so doubling the length to $2l$ forces the area to halve to $A/2$. The new resistance becomes $R^{\prime }=\rho \frac{2l}{A/2}=4\rho \frac{l}{A}=4R$. More generally, at constant volume the area can be written as $A=V/l$, so resistance becomes $R=\rho l^2/V$, meaning resistance scales as the square of the length; thus a length factor of 2 gives a resistance factor of 4. The option 2R accounts only for the doubling of length and forgets that the area shrinks to half simultaneously, so it underestimates the change. The option R/2 wrongly treats the cross-sectional area as increasing, which cannot happen when volume is fixed and length grows. The option R/4 inverts the correct dependence entirely, as if lengthening the wire lowered its resistance. A consistency check confirms the reasoning: since $R\propto l^2$ under constant volume, $2^2=4$, and physically the resistance can only increase when a wire is made both longer and thinner, so the fourfold rise in magnitude is entirely sensible.