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Power Delivered and Heat Dissipated in a Circuit

A battery of ...

Question

A battery of emf E and internal resistance r is connected to a resistor of resistance $r_{1}$ and Q joules of heat is produced in a certain time t. When the same battery is connected to another resistor of resistance $r_{2}$ the same quantity of heat is produced in the same time t the value of r is

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Solution

The correct option is D
In the first case the current in the circuit is
$l_{1} = \frac{E}{r_{1} + r}$
$Q_{1} = l_{2} r_{1} t = (\frac{E}{r_{1} + r})^{2} \times r_{1} \times t$
In the second case
$Q_{2} = (\frac{E}{r_{1} + r})^{2} \times r_{2} \times t$
equating above equations we get
$\frac{r_{1}}{(r_{1} + r)^{2}} = \frac{r_{2}}{(r_{2} + r)^{2}}$
$r_{1} (r_{2} + r)^{2} = r_{2} (r_{1} + r)^{2}$
$r_{1} (r_{2}^{2} + 2 r r_{2} + r^{2}) = r_{2} (r_{1}^{2} + 2 r r_{1} + r^{2})$
$r^{2} (r_{1} - r_{2}) = r_{1} r_{2} (r_{1} - r_{2})$
$r = \sqrt{r_{1} r_{2}}$

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