A) \[\sqrt{{{R}_{1}}{{R}_{2}}}\]
B) \[\sqrt{\frac{{{R}_{1}}}{{{R}_{2}}}}\]
C) \[\frac{{{R}_{1}}-{{R}_{2}}}{2}\]
D) \[\frac{{{R}_{1}}+{{R}_{2}}}{2}\]
Correct Answer: A
Solution :
Power dissipated\[={{i}^{2}}R={{\left( \frac{E}{R+r} \right)}^{2}}R\] \ \[{{\left( \frac{E}{{{R}_{1}}+r} \right)}^{2}}{{R}_{1}}={{\left( \frac{E}{{{R}_{2}}+r} \right)}^{2}}{{R}_{2}}\] Þ \[{{R}_{1}}(R_{2}^{2}+{{r}_{2}}+2{{R}_{2}}r)={{R}_{2}}(R_{1}^{2}+{{r}^{2}}+2{{R}_{1}}r)\] Þ \[R_{2}^{2}{{R}_{1}}+{{R}_{1}}{{r}^{2}}+2{{R}_{2}}r=R_{1}^{2}{{R}_{2}}+{{R}_{2}}{{r}^{2}}+2{{R}_{1}}{{R}_{2}}r\] Þ \[({{R}_{1}}-{{R}_{2}}){{r}^{2}}=({{R}_{1}}-{{R}_{2}}){{r}^{2}}=({{R}_{1}}-{{R}_{2}}){{R}_{1}}{{R}_{2}}\] Þ \[r=\sqrt{{{R}_{1}}{{R}_{2}}}\]
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