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The ratio of the resistance of conductor at temperature 15°C to its resistance at temperature 37.5°C is 4:5 . The temperature coefficient of resistance of the conductor is -
- a)1/20 ºC-1
- b)1/50 ºC-1
- c)1/80 ºC-1
- d)1/75 ºC-1
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The ratio of the resistance of conductor at temperature 15°C to its re...
Understanding Resistance and Temperature Relationship
The resistance of a conductor changes with temperature, which can be described by the formula:
R(T) = R0(1 + α(T - T0))
Where:
- R(T) = Resistance at temperature T
- R0 = Resistance at a reference temperature T0
- α = Temperature coefficient of resistance
- T = New temperature
- T0 = Reference temperature
Given Data
- Resistance ratio at 15°C and 37.5°C is 4:5.
- Let R1 = Resistance at 15°C and R2 = Resistance at 37.5°C.
From the ratio, we can express it as:
R1/R2 = 4/5
This leads to:
R2 = (5/4) R1
Applying the Resistance Formula
Using the resistance formula for both temperatures:
R1 = R0(1 + α(15 - T0))
R2 = R0(1 + α(37.5 - T0))
Substituting R2 into the resistance ratio:
R0(1 + α(15 - T0)) / R0(1 + α(37.5 - T0)) = 4/5
This simplifies to:
(1 + α(15 - T0)) / (1 + α(37.5 - T0)) = 4/5
Solving for Temperature Coefficient (α)
Cross-multiplying gives:
5(1 + α(15 - T0)) = 4(1 + α(37.5 - T0))
Expanding both sides:
5 + 5α(15 - T0) = 4 + 4α(37.5 - T0)
Rearranging leads to:
α(5(15 - T0) - 4(37.5 - T0)) = -1
Solving for α leads us to find:
α = -1 / (5(15 - T0) - 4(37.5 - T0))
When T0 is taken as 0°C, we find:
α = -1/75 °C^-1
Conclusion
The temperature coefficient of resistance of the conductor is indeed -1/75 °C^-1, confirming that option 'D' is the correct answer.
The resistance of a conductor changes with temperature, which can be described by the formula:
R(T) = R0(1 + α(T - T0))
Where:
- R(T) = Resistance at temperature T
- R0 = Resistance at a reference temperature T0
- α = Temperature coefficient of resistance
- T = New temperature
- T0 = Reference temperature
Given Data
- Resistance ratio at 15°C and 37.5°C is 4:5.
- Let R1 = Resistance at 15°C and R2 = Resistance at 37.5°C.
From the ratio, we can express it as:
R1/R2 = 4/5
This leads to:
R2 = (5/4) R1
Applying the Resistance Formula
Using the resistance formula for both temperatures:
R1 = R0(1 + α(15 - T0))
R2 = R0(1 + α(37.5 - T0))
Substituting R2 into the resistance ratio:
R0(1 + α(15 - T0)) / R0(1 + α(37.5 - T0)) = 4/5
This simplifies to:
(1 + α(15 - T0)) / (1 + α(37.5 - T0)) = 4/5
Solving for Temperature Coefficient (α)
Cross-multiplying gives:
5(1 + α(15 - T0)) = 4(1 + α(37.5 - T0))
Expanding both sides:
5 + 5α(15 - T0) = 4 + 4α(37.5 - T0)
Rearranging leads to:
α(5(15 - T0) - 4(37.5 - T0)) = -1
Solving for α leads us to find:
α = -1 / (5(15 - T0) - 4(37.5 - T0))
When T0 is taken as 0°C, we find:
α = -1/75 °C^-1
Conclusion
The temperature coefficient of resistance of the conductor is indeed -1/75 °C^-1, confirming that option 'D' is the correct answer.
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Community Answer
The ratio of the resistance of conductor at temperature 15°C to its re...
The resistance at temperature T is given by the formula:
RT = R0 (1 + αT)
Given, ratio of resistances at 15°C and 37.5°C is:
R15 / R37.5 = 4/5
Using the formula:
(R0(1 + α × 15)) / (R0(1 + α × 37.5)) = 4/5
Therefore,
(1 + 15α) / (1 + 37.5α) = 4/5
Cross-multiplying,
5(1 + 15α) = 4(1 + 37.5α)
5 + 75α = 4 + 150α
75α - 150α = 4 - 5
-75α = -1
α = 1/75 °C-1
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