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

For parallel combination of resistors, net resistance is, 1R = 1R_1 + 1R_2 R = R_1 R_2R_1 + R_2 Differentiating on both sides, dRR^2 = dR_1R^2_1 + ( d ...

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Standard XII

Physics

Idea of Temperature

Two resistors...

Question

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

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

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

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Solution

The correct option is D $\frac{R_{1} α_{2} + R_{2} α_{1}}{R_{1} + R_{2}}$
For parallel combination of resistors, net resistance is,

$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$

$R = \frac{R_{1} R_{2}}{R_{1} + R_{2}}$

Differentiating on both sides,

$- \frac{d R}{R^{2}} = - \frac{d R_{1}}{R_{1}^{2}} + (- \frac{d R_{2}}{R_{2}^{2}})$

$\frac{d R}{R} = R [\frac{R_{1} α_{1} t}{R_{1}^{2}} + \frac{R_{2} α_{2} t}{R_{2}^{2}}]$

$\frac{d R}{R t} = \frac{R_{1} R_{2}}{R_{1} + R_{2}} [\frac{R_{2} α_{1} + R_{1} α_{2}}{R_{1} R_{2}}]$

$∴ α = \frac{d R}{R t} = \frac{R_{2} α_{1} + R_{1} α_{2}}{R_{1} + R_{2}}$

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

The correct option is D R1α2+R2α1R1+R2For parallel combination of resistors, net resistance is, 1R=1R1+1R2 R=R1R2R1+R2 Differentiating on both sides, −dRR2=−dR1R21+(−dR2R22) dRR=R[R1α1tR21+R2α2tR22] dRRt=R1R2R1+R2[R2α1+R1α2R1R2] ∴α=dRRt=R2α1+R1α2R1+R2

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