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.