Power generated in \(R\)

\(P=i^{2} R\)

\(P=\frac{E^{2} R}{(r+R)^{2}}\)

For maximum power \(\frac{\mathrm{dP}}{\mathrm{dR}}=0\) 

\(E^{2}\left[\frac{(r+R)^{2} \times 1-R \times 2(r+R)}{(r+R)^{4}}\right]=0\)

\(\Rightarrow r=R\)