Two resistance R1 and R2 are made of different materials- The temperature coefficient of the material of R1 is - and that of the material of R2 is - The resistance of the series combination of R1 and R2 does not change with temperature- then the ratio of resistances of the two wires at 0-C will be-

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Resistance and Resistors

Two resistanc...

Question

Two resistance $R_{1}$ and $R_{2}$ are made of different materials. The temperature coefficient of the material of $R_{1}$ is $α$ and that of the material of $R_{2}$ is $- β$ . The resistance of the series combination of $R_{1}$ and $R_{2}$ does not change with temperature, then the ratio of resistances of the two wires at $0^{\circ} C$ will be:

\frac{2 α}{β}

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\frac{α + β}{α - β}

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\frac{α^{2} + β^{2}}{α β}

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\frac{β}{α}

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Solution

The correct option is D $\frac{β}{α}$
Resistance of wire as temperature changes is
$R_{1} = R_{01} (1 + α Δ T)$
$R_{2} = R_{02} (1 - β . Δ T)$

$R_{series} = R_{1} + R_{2} = R_{01} + (R_{01} α - R_{02} β) Δ T + R_{02}$
$R_{series}$ is independent of temperature, if
$R_{01} α - R_{02} β = 0 \Rightarrow \frac{R_{01}}{R_{02}} = \frac{β}{α}$

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The correct option is D βαResistance of wire as temperature changes is R1=R01(1+αΔT) R2=R02(1−β.ΔT) Rseries=R1+R2=R01+(R01α−R02β)ΔT+R02 Rseries is independent of temperature, if R01α−R02β=0⇒R01R02=βα