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Non-Ideal Battery

A current sou...

Question

A current source drives a current in a coil of resistance $R_{1}$ for a time $t$ . The same source drives current in another coil of resistance $R_{2}$ for same time. If heat generated is same, find internal resistance of source.

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

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R_{1} + R_{2}

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zero

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

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Solution

The correct option is B $\sqrt{R_{1} R_{2}}$
Let internal resistance of source $= r$
Current in coil of resistance $R_{1}$ is $I_{1} = \frac{V}{R_{1} + r}$ where $V =$ voltage of source.
Current in coil of resistance $R_{2}$ is $I_{2} = \frac{V}{R_{2} + r}$
As heat generated in both coil is same, so $H_{1} = H_{2} \Rightarrow I_{1}^{2} R_{1} t = I_{2}^{2} R_{2} t$
$(\frac{V}{R_{1} + r})^{2} R_{1} = (\frac{V}{R_{2} + r})^{2} R_{2}$
$R_{1} (R_{2} + r)^{2} = R_{2} (R_{1} + r)^{2}$
$R_{1} R_{2}^{2} + 2 R_{1} R_{2} r + R_{1} r^{2} = R_{1}^{2} R_{2} + 2 R_{1} R_{2} r + R_{2} r^{2}$
$R_{1} R_{2}^{2} + R_{1} r^{2} - R_{1}^{2} R_{2} - R_{2} r^{2} = 0$

$R_{1} r^{2} - R_{2} r^{2} = R_{1}^{2} R_{2} - R_{1} R_{2}^{2}$
$r^{2} (R_{1} - R_{2}) = R_{1} R_{2} (R_{1} - R_{2})$
$r^{2} = R_{1} R_{2}$

Thus, internal resistance $r = \sqrt{R_{1} R_{2}}$

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