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If two resistive materials with temperature coefficients of resistance 0.004 and 0.0004 per °C are connected in series, what is the overall temperature coefficient for the series combination?

Difficulty: Easy

Correct Answer: Data insufficient for a unique value

Explanation:


Introduction / Context:
The temperature coefficient of resistance (TCR) describes how resistance changes with temperature: R(T) = R_0 [1 + α ΔT]. For series combinations, the effective α depends on each element’s α and its share of the total resistance at the reference temperature. Without knowing the individual resistances (or a ratio), you cannot compute a unique combined TCR.



Given Data / Assumptions:

  • Two materials with α₁ = 0.004/°C and α₂ = 0.0004/°C.
  • They are connected in series.
  • No information is provided about R₁ and R₂ at the reference temperature.


Concept / Approach:
For series elements, R_total = R₁ + R₂ and the effective coefficient is α_series = (α₁ R₁ + α₂ R₂) / (R₁ + R₂). This weighted average depends on the resistance ratio R₁:R₂. Hence different ratios produce different α_series values.



Step-by-Step Solution:

Write R_total(T) = R₁(1 + α₁ ΔT) + R₂(1 + α₂ ΔT).Factor: R_total(T) = (R₁ + R₂) [1 + ((α₁ R₁ + α₂ R₂)/(R₁ + R₂)) ΔT].Thus α_series = (α₁ R₁ + α₂ R₂)/(R₁ + R₂), which is not uniquely determined by α₁ and α₂ alone.


Verification / Alternative check:

Example 1 (R₁ = R₂): α_series = (0.004 + 0.0004)/2 = 0.0022.Example 2 (R₁ = 9 R₂): α_series ≈ (0.0049 + 0.00041)/10 = 0.00364.


Why Other Options Are Wrong:

Fixed numbers like 0.08, 0.04, 0.001, or 0.0001 cannot be derived without R₁ and R₂.


Common Pitfalls:

Assuming a simple arithmetic mean; overlooking the resistance weighting in series combinations.


Final Answer:

Data insufficient for a unique value
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