$A$ potential difference $V$ is applied across a conductor of length $L$ and diameter $D$. How is the resistance $R$ of the conductor affected,when in turn $(i)$ $V$ is halved,$(ii)$ $L$ is halved,and $(iii)$ $D$ is doubled? Justify your answer in each case.

(A) The resistance $R$ of a conductor is given by the formula $R = \rho \frac{L}{A}$,where $\rho$ is the resistivity,$L$ is the length,and $A$ is the cross-sectional area. The area $A = \pi r^2 = \pi (D/2)^2 = \frac{\pi D^2}{4}$.
$(i)$ When $V$ is halved: The resistance $R$ depends only on the physical dimensions and material properties of the conductor. Therefore,$R$ remains unchanged.
$(ii)$ When $L$ is halved: Since $R \propto L$,if the length $L$ is halved,the resistance $R$ also becomes halved.
$(iii)$ When $D$ is doubled: Since $R \propto \frac{1}{A}$ and $A \propto D^2$,we have $R \propto \frac{1}{D^2}$. If $D$ is doubled,$R$ becomes $\frac{1}{(2)^2} = \frac{1}{4}$ of its original value.