\(\int_{0}^{R} d R=k \int_{0}^{1} \frac{d \ell}{\sqrt{\ell}}\)

\(\mathrm{R}=2 \mathrm{k}\) resistance of wire \(\mathrm{AB}\)

Again, \(\int_{0}^{R / 2} \mathrm{d} \mathrm{R}=\mathrm{k} \int_{0}^{L} \frac{\mathrm{d} \ell}{\sqrt{\ell}} \quad \mathrm{L} \rightarrow\) Length \(\mathrm{AP}\)

\(\frac{\mathrm{R}}{2}=\mathrm{k} 2 \mathrm{L}^{1 / 2} \quad ; \quad \mathrm{k}=\mathrm{k} 2 \mathrm{L}^{1 / 2}\)

\(\Rightarrow \quad \mathrm{L}=\frac{1}{4} \,\mathrm{m}=0.25\, \mathrm{m}\)