First slide

Electrical power

Question

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

Moderate

A

$R_{1} + R_{2}$
B

$\frac{R_{1} + R_{2}}{2}$
C

$\sqrt{{R_{1}}^{2} + {R_{2}}^{2}}$
D

$\sqrt{R_{1} R_{2}}$

Solution

${(\frac{e}{R_{1} + r})}^{2} R_{1} = {(\frac{e}{R_{2} + r})}^{2} R_{2}$
$R_{1} (R_{2}^{2} + r^{2} + 2 R_{2} r) = R_{2} (R_{1}^{2} + r^{2} + 2 R_{1} r)$
$R_{1} R_{2}^{2} + R_{1} r^{2} + 2 R_{1} R_{2} r = R_{2} R_{1}^{2} + R_{2} r^{2} + 2 R_{1} R_{2} r$
$(R_{1} - R_{2}) r^{2} = R_{1} R_{2} (R_{1} - R_{2})$
$r = \sqrt{R_{1} R_{2}}$

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