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A circuit shown in the figure has resistances $20 Ω and 30 Ω$ . At what value of resistance R_x will the thermal power generated in it be pracitcally independent of small variations of that resistance? The voltage between points A and B is supposed to be constant in this case

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$12 Ω$

$3 Ω$

$20 Ω$

$7 Ω$

answer is C.

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Detailed Solution

Question Image From shown circuit

$I = \frac{E}{20 + \frac{30 R_{X}}{30 + R_{X}}} . . . . . . . (1)$ and $I_{1} = \frac{I .30}{30 + R_{X}} . . . . . . . (2)$

From (1)&(2), $[I_{1} = \frac{3 E}{60 + 5 R_{X}}]$

Power across $R_{x} is P = I_{1}^{2} R = \frac{9 E^{2}}{{(60 + 5 R_{x})}^{2}} R_{x}$

P will be constant when $\frac{dP}{{dR}_{x}} = 0$

$9 E^{2} \frac{d}{{dR}_{x}} [{(60 + 5 R_{x})}^{- 2} R_{x}] = 0$

$\Rightarrow {(60 + 5 R_{x})}^{- 2} + R_{x} {(60 + 5 R_{x})}^{- 3} (- 2) (5) = 0$

$1 - 10 R_{x} {(60 + 5 R_{x})}^{- 1} = 0$

$\Rightarrow \frac{10 R_{x}}{60 + 5 R_{x}} = 1 \Rightarrow R_{x} = 12 Ω .$

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