Question

A cell sends a current through a resistance $R$ for time $t$. Now the same cell sends current through another resistance $r$ for the same time. If same amount of heat is developed in both the resistance, then the intemal resistance of cell is

• A. $\frac{(R+r)}{2}$
• B. $\frac{(R-r)}{2}$
• C. $\frac{(R r)}{2}$
• D. $\sqrt{(R r)}$

Answer 1

Answer: (D)

$H_{1}=\frac{E^{2} R}{\left(R+r_{1}\right)^{2}}$
$H_{2}=\frac{E^{2} R}{\left(r+r_{1}\right)^{2}}$
$\frac{R}{\left(R+r_{1}\right)^{2}}=\frac{r}{\left(r+r_{1}\right)^{2}}$
$\frac{r+r_{1}}{R+r_{1}}=\frac{\sqrt{r}}{\sqrt{R}}$
$\sqrt{R} \,r_{1}+r \sqrt{R}$
$=\sqrt{r}\, r_{1}+R \sqrt{r}$
$r_{1}(\sqrt{R}-\sqrt{r})$
$=\sqrt{R r}(\sqrt{R}-\sqrt{r})$
$r_{1}=\sqrt{R r}$