a
Given, ${R_1} = 100\,\Omega ,{r^\prime } = r/2,{R_2} = ?$
Resistivity of wire, $\mathrm{R}=\frac{\rho l}{\mathrm{A}}$
$\because$ Area $\times$ length $=$ volume
Hence, $R=\frac{\rho V}{A^{2}}$
since, $\rho \rightarrow$ constant, $V \rightarrow$ constant
$\mathrm{R} \propto \frac{1}{\mathrm{A}^{2}}$
or $\quad \mathrm{R} \propto \frac{1}{\mathrm{r}^{4}} \quad \because \quad \mathrm{A}=\pi \mathrm{r}^{2}$
$\frac{\mathrm{R}_{2}}{\mathrm{R}_{1}}=16 \Rightarrow \mathrm{R}_{2}=16 \times 100=1600 \,\Omega,$
Resistance of new wire.