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Q.

For ensuring dissipation of same energy in all three resistors (R₁, R₂, R₃) connected as shown in figure, their values must be related as

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a

$R_{2} = R_{3} and R_{1} = 2 R_{2}$

b

$R_{2} = R_{3} and R_{1} = (\frac{1}{4}) R_{2}$

c

$R_{1} = R_{2} + R_{3}$

d

$R_{1} = R_{2} = R_{3}$

answer is C.

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Detailed Solution

$Q_{1} = Q_{2} = Q_{3} i_{1}^{2} R_{1} = i_{2}^{2} R_{2} = i_{3}^{2} R_{3}$

$i_{1}^{2} R_{1} = {(\frac{i_{1} R_{3}}{R_{2} + R_{3}})}^{2} R_{2} = {(\frac{i_{1} R_{2}}{R_{2} + R_{3}})}^{2} R_{3}$

$R_{1} = \frac{R_{2} R_{3}^{2}}{{(R_{2} + R_{3})}^{2}} = \frac{R_{2}^{2} R_{3}}{{(R_{2} + R_{3})}^{2}}$

$∴ R_{1} {(R_{2} + R_{3})}^{2} = R_{2} R_{3}^{2} = R_{3} R_{2}^{2}$

$R_{2} = R_{3}, R_{1} = \frac{R_{2}}{4}$

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For ensuring dissipation of same energy in all three resistors (R1, R2, R3) connected as shown in figure, their values must be related as