Answer:
-17.04 [tex]^0 C[/tex]
Explanation:
The resistance of an object varies with temperature and is given by the formula:
[tex]R=R_0(1+\alpha (T - T_0)[/tex]
Where R = resistance at temperature T
[tex]R_0[/tex] = resistance at reference temperature [tex]T_0[/tex]
[tex]\alpha[/tex] = Temperature coefficient of resistance
T = temperature in degree Celsius
[tex]T_0[/tex] = reference temperature.
For the copper wire:
R = 0.500(1 + 0.004041(20.0 - [tex]T_0[/tex])
For the iron wire:
R = 0.525(1 + 0.005671(20.0 - [tex]T_0[/tex])
When the resistance are equal for the two elements:
0.500(1 + 0.004041(20.0 - [tex]T_0[/tex]) = 0.525(1 + 0.005671(20.0 - [tex]T_0[/tex])
0.500(1 + 0.08082 - 0.004041[tex]T_0[/tex]) = 0.525(1 + 0.11342 - 0.005671[tex]T_0[/tex])
0.000956775[tex]T_0[/tex] = -0.0163
[tex]T_0[/tex] = -17.04
The temperature of equal resistance is -17.04 [tex]^0 C[/tex]
At -6 Degree Celsius the two materials (copper wire and iron wire) have the same resistance.
Given data:
The temperature is, T' = 20.0ºC.
The resistance of copper wire is, R = 0.500 Ω .
The resistance of iron wire is, R' = 0.525 Ω.
The given problem is based on the effect of resistance due to temperature. Resistance has proportional effect due to temperature as resistance increase with increasing the temperature. So, the relation is given as,
For copper wire:
[tex]R"=R(1+ \alpha(T-T'}))[/tex]
Here, T is the final temperature and [tex]\alpha[/tex] is the temperature coefficient and for copper its value is 0.004021.
Then,
[tex]R"=0.500 \times(1+ 0.004041(T-20})) ...........................................................(1)[/tex]
And for iron, the temperature coefficient is 0.005671. Then,
[tex]R"=0.525 \times(1+ 0.005671(T-20})) ...........................................................(2)[/tex]
For same resistance, compare equation (1) and (2) as,
[tex]0.500 \times(1+ 0.004041(T-20})) =0.525 \times(1+ 0.005671(T-20}))\\\\0.500 +(2.02 \times 10^{-3} )T - 0.040 = 0.525 + (2.97 \times 10^{-3})T -0.059\\\\-6 \times 10^{-3} = 9.5 \times 10^{-4} \times T\\\\T = -6.31 ^{\circ}\rm C[/tex]
Thus, we can conclude that at -6 Degree Celsius the two materials have the same resistance.
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