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 resistance of the two wires at 0-C will be-

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EMF and EMF Devices

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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 resistance of the two wires at $0^{\circ} C$ will be:

\frac{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 $\frac{β}{α}$
Temperature dependence of resistance is,

$R_{t} = R (1 + α Δ T)$

Let, temperature dependence resistance of $R_{1}$ and $R_{2}$ be $R_{1 t}$ and $R_{2 t}$ respectively.

$R_{1 t} = R_{1} (1 + α Δ T)$

$R_{2 t} = R_{2} (1 - β Δ T)$

When they connected in series, the equivalent resistance of the combination is,

$R_{e q} = R_{1 t} + R_{2 t}$

$= (R_{1} + R_{2}) + Δ T (R_{1} α - R_{2} β)$

Since, it is temperature independent,

$∴ Δ T (R_{1} α - R_{2} β) = 0$

$\Rightarrow R_{1} α = R_{2} β$

$\Rightarrow \frac{R_{1}}{R_{2}} = \frac{β}{α}$

 Hence, $(D)$ is the correct answer.

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