A copper wire has a resistance of 0.425 at 20.0°C, and an iron wire has a resistance of 0.455 at the same temperature. At what temperature are their resistances equal?
I got the solution:
copper=3.9x10^-3
iron=5.0x10^-3
R(1+α∆T)=R(1+α∆T)
0.425(1+0.0039∆T)=0.455(1+0.005∆T)
∆T=-48.6°C
T=20°C-48.6°C=-28.6°C
Given the temperature coefficients of resistance (α) of each metal at 20°C
copper = 6.8×10^-3[/°C]
iron = 6.51×10^-3[/°C]
Use: R(T+∆T) = R(T) + α R(T) ∆T
Find: (20°C + ∆T) such that:
0.425+0.425×0.0068 ∆T = 0.444+0.425×0.00651 ∆T
the length of copper and iron wire is 500m at 0°c. determine the difference in their length at 30°c
To find the temperature at which the resistances of the copper wire and iron wire are equal, we can use the formula for temperature dependence of resistance:
R_t = R_0 * (1 + α * (T - T_0))
Where:
R_t is the resistance at temperature T,
R_0 is the resistance at temperature T_0,
α is the temperature coefficient of resistance, and
T_0 is the reference temperature.
First, let's determine the temperature coefficients of resistance for copper and iron.
The temperature coefficient of resistance (α) for copper is approximately 0.00393 per °C.
The temperature coefficient of resistance (α) for iron is approximately 0.00651 per °C.
Now, we set up the equation:
0.425 * (1 + 0.00393 * (T - 20.0)) = 0.455 * (1 + 0.00651 * (T - 20.0))
We simplify the equation:
0.425 + 0.00167 * (T - 20.0) = 0.455 + 0.0029595 * (T - 20.0)
0.00167T - 0.0334 = 0.0029595T - 0.004783
Combining like terms:
0.00167T - 0.0029595T = 0.0334 - 0.004783
-0.0012795T = 0.028617
Dividing by -0.0012795:
T = 22.38°C
Therefore, at approximately 22.38°C, the resistances of the copper wire and iron wire are equal.