Question

A cell sends a current through a resistance$R$ for time $t$; next the same cell sends current through another resistance $r$ for the heat is developed in both the resistance, then the internal resistance of the cell is:

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

Answer 1

The correct option is C $\sqrt{Rr}$

$H_R=I_R^{2}Rt$ $[$ Heat produced in $R]$

$H_r=I_r^{2}rt$ $[$ Heat Produced in $r]$

According to question

$H_R=H_r$

$I_R^{2}Rt=I_r^{2}rt$

$\because I_R=\frac{E}{R+r_1}$ $[r_1=$ Internal resistance of cell $]$

$I_r=\frac{E}{r+r_1}$ $E=$ Potential of cell $]$

$\therefore \frac{E^2}{(R+r_1{)}^2}.R=\frac{E^2}{(r+r_1{)}^2}.r$

$\sqrt{\frac{R}{r}}=\frac{R+r_1}{r+r_1}$

$\sqrt{R}r+\sqrt{R}{r}_1=\sqrt{r}R+\sqrt{r}{r}_1$

$\sqrt{R}r-\sqrt{r}R=r_1(\sqrt{r}-\sqrt{R})$

$r_1=\frac{\sqrt{Rr}(\sqrt{r}-\sqrt{R})}{(\sqrt{r}-\sqrt{R)}}$

$r_1=\sqrt{Rr}$