 Electrical power
Question

# A cell dissipates the same power across each of two resistors ${\mathrm{R}}_{1}$ and ${\mathrm{R}}_{2}$ when connected separately. The internal resistance of cell is

Moderate
Solution

## ${\left(\frac{\mathrm{e}}{{\mathrm{R}}_{1}+\mathrm{r}}\right)}^{2}{\mathrm{R}}_{1}={\left(\frac{\mathrm{e}}{{\mathrm{R}}_{2}+\mathrm{r}}\right)}^{2}{\mathrm{R}}_{2}$${\text{R}}_{1}\left({\mathrm{R}}_{2}^{2}+{\mathrm{r}}^{2}+2{\mathrm{R}}_{2}\mathrm{r}\right)={\mathrm{R}}_{2}\left({\mathrm{R}}_{1}^{2}+{\mathrm{r}}^{2}+2{\mathrm{R}}_{1}\mathrm{r}\right)$${\mathrm{R}}_{1}{\mathrm{R}}_{2}^{2}+{\mathrm{R}}_{1}{\mathrm{r}}^{2}+2{\mathrm{R}}_{1}{\mathrm{R}}_{2}\mathrm{r}={\mathrm{R}}_{2}{\mathrm{R}}_{1}^{2}+{\mathrm{R}}_{2}{\mathrm{r}}^{2}+2{\mathrm{R}}_{1}{\mathrm{R}}_{2}\mathrm{r}$$\left({\mathrm{R}}_{1}-{\mathrm{R}}_{2}\right){\mathrm{r}}^{2}={\mathrm{R}}_{1}{\mathrm{R}}_{2}\left({\mathrm{R}}_{1}-{\mathrm{R}}_{2}\right)$$\mathrm{r}=\sqrt{{\mathrm{R}}_{1}{\mathrm{R}}_{2}}$

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