If the resistance of an aluminum wire at room temperature (20°C) is 100 ohm, at what temperature will its resistance increase to 120 ohm?
To solve this problem, we can use the formula for calculating the temperature coefficient of resistance (α) for aluminum:
Rt = Ro * (1 + α * (t - To))
where:
- Rt is the resistance at temperature t
- Ro is the resistance at the reference temperature To
- α is the temperature coefficient of resistance for aluminum
To find the temperature at which the resistance increases to 120 ohm, we will set Rt = 120 ohm, Ro = 100 ohm, and solve for t.
120 = 100 * (1 + α * (t - 20))
Dividing both sides of the equation by 100:
1.2 = 1 + α * (t - 20)
Rearranging the equation:
α * (t - 20) = 1.2 - 1
α * (t - 20) = 0.2
Now, we need to find the temperature coefficient of resistance (α) for aluminum. The temperature coefficient of resistance, α, is typically given in units of ohm per degree Celsius (Ω/°C). For aluminum, the average temperature coefficient of resistance is approximately 0.0039 Ω/°C.
So, α = 0.0039 Ω/°C.
0.0039 * (t - 20) = 0.2
Dividing both sides of the equation by 0.0039:
t - 20 = 0.2 / 0.0039
t - 20 = 51.28
Adding 20 to both sides of the equation:
t = 51.28 + 20
t ≈ 71.28
Therefore, the resistance of the aluminum wire will increase to 120 ohm at a temperature of approximately 71.28°C.