Question

A cell of constant emf first connected to a resistance $R_1$ and then connected to a resistance $R_2$ . If power delivered in both cases is same, then the internal resistance of the cell is

• A. $\sqrt{R_1{R}_2}$
• B. $\sqrt{\frac{R_1}{R_2}}$
• C. $\frac{R_1-R_2}{2}$
• D. $\frac{R_1+R_2}{2}$

Answer 1

Answer: (A)

Current given by cell $I=\frac{E}{R+r}$
Power delivered in second case $P_2=I^2{R}_2$
$={\left(\frac{E}{R_2+r}\right)}^2{R}_2$
Power delivered is same in the both cases
${\left(\frac{E}{R_2+r}\right)}^2{R}_1={\left(\frac{E}{R_2+r}\right)}^2$
$R_1(R_2^2+r^2+2R_{2}r)=R_2(R_1^2{r}^2+r^{2}2R_{1}r)$
$R_1{R}_2^2+R_1{r}^2+2R_1{R}_{2}r=R_2{R}_1^2{r}^2+R_2{r}^2+2R_1{R}_{2}r$
$R_1{R}_2^2-R_2{R}_1^2=R_2{R}^2-R_1{r}^2$
$R_1{R}_2(R_2-R_1)=R_2{r}^2-R_1{r}^2$
$r=\sqrt{R_1{R}_2}$