question_answer

Two cells, each of emf£ and internal resistance r are connected in parallel across a resistance R. The power delivered to R is maximum when

A)  \[r=R\]                

B)   \[R=\frac{r}{2}\]

C)  \[R=2\,r\]                          

D)  \[R=0\]

Correct Answer: B

Solution :

 Let I be the current through the circuit, then effective resistance \[R=R+\frac{r}{2}\] and        \[I=\frac{E}{R+\frac{r}{2}}\] From Joules law \[P={{I}^{2}}R\] \[\Rightarrow \]               \[P={{\left( \frac{E}{R+\frac{r}{2}} \right)}^{2}}R\] \[\Rightarrow \]               \[P=\frac{4{{E}^{2}}R}{{{(r+2R)}^{2}}}\] For maximum power \[\frac{dP}{dR}=0\] \[\therefore \]  \[\frac{dP}{dR}=0=\frac{4r{{E}^{2}}-8{{E}^{2}}R}{{{(r+2R)}^{3}}}\]. \[\Rightarrow \]               \[R=\frac{r}{2}\]