Two resistors with temperature coefficients of resistance -1 and -2 have resistances R1 and R2 at 0- C- What is the temperature coefficient of the compound resistor consisting of the two resistors connected in parallel-

When connected in parallel, 1/R = 1/R_1 + 1/R_2 Differentiating both sides, dR/R^2 = dR_1/R_1^2 + dR_2/R_2^2 ⇒α R dT/R^2 = α_1R_1dT/R_1^2 + α_2 R_2dT/R_2^2 ⇒ ...

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Byju's Answer

Standard XII

Physics

Idea of Temperature

Two resistors...

Question

Two resistors with temperature coefficients of resistance $α_{1}$ and $α_{2}$ have resistances $R_{1}$ and $R_{2}$ at $0^{\circ} C$ . What is the temperature coefficient of the compound resistor consisting of the two resistors connected in parallel?

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

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$\frac{α_{1} + α_{2}}{2}$

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$\frac{2 α_{1} + α_{2}}{α_{1} + α_{2}}$

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

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Solution

The correct option is D
$\frac{α_{1} R_{2} + α_{2} R_{1}}{R_{1} + R_{2}}$

When connected in parallel,
$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$
Differentiating both sides,
$\frac{- d R}{R^{2}} = \frac{- d R_{1}}{R_{1}^{2}} + \frac{- d R_{2}}{R_{2}^{2}} \Rightarrow \frac{α R d T}{R^{2}} = \frac{α_{1} R_{1} d T}{R_{1}^{2}} + \frac{α_{2} R_{2} d T}{R_{2}^{2}} \Rightarrow \frac{α}{R} = \frac{α_{1}}{R_{1}} + \frac{α_{2}}{R_{2}} \Rightarrow α = R (\frac{α_{1}}{R_{1}} + \frac{α_{2}}{R_{2}}) = \frac{R_{1} R_{2}}{R_{1} + R_{2}} (\frac{α_{1} R_{2} + α_{2} R_{1}}{R_{1} R_{2}}) ∴ α = \frac{α_{1} R_{2} + α_{2} R_{1}}{R_{1} + R_{2}}$

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Two resistors with temperature coefficients of resistance α1 and α2 have resistances R1 and R2 at 0∘C. What is the temperature coefficient of the compound resistor consisting of the two resistors connected in parallel?

The correct option is D α1R2+α2R1R1+R2 When connected in parallel, 1R=1R1+1R2 Differentiating both sides, −dRR2=−dR1R21+−dR2R22⇒αRdTR2=α1R1dTR21+α2R2dTR22⇒αR=α1R1+α2R2⇒α=R(α1R1+α2R2)=R1R2R1+R2(α1R2+α2R1R1R2)∴α=α1R2+α2R1R1+R2

Two resistors with temperature coefficients of resistance $α_{1}$ and $α_{2}$ have resistances $R_{1}$ and $R_{2}$ at $0^{\circ} C$ . What is the temperature coefficient of the compound resistor consisting of the two resistors connected in parallel?