Step-by-step explanation:
To find the rate of energy dissipated by the wire when the temperature is 741ºC, we can use the concept of temperature dependence of resistance in a conductor.
The formula to calculate the power (P) dissipated by a resistor with temperature coefficient of resistivity (α) is given by:
P2 = P1 * [(R2 / R1)^2]
where:
P1 = Power dissipated by the resistor at temperature T1
P2 = Power dissipated by the resistor at temperature T2
R1 = Resistance of the resistor at temperature T1
R2 = Resistance of the resistor at temperature T2
In this case, we are given that the power (P1) dissipated by the wire is 89.0W when the temperature (T1) is -37.0ºC. We need to find the power (P2) when the temperature (T2) is 741ºC.
Step 1: Find the resistance at T1:
We can use the formula for power to calculate the resistance at T1:
P1 = I^2 * R1
R1 = P1 / I^2
Step 2: Calculate the resistance at T2:
Using the formula for temperature dependence of resistance:
R2 = R1 * [1 + α * (T2 - T1)]
Step 3: Find the power at T2:
Using the formula for power, with the calculated R2 value:
P2 = I^2 * R2
Now, let's calculate the values:
Given data:
P1 = 89.0W (power dissipated at T1)
T1 = -37.0ºC (initial temperature)
T2 = 741ºC (final temperature)
α = 0.00350(∘)−1 (temperature coefficient of resistivity)
Step 1:
R1 = P1 / I^2
R1 = 89.0 / I^2
Step 2:
R2 = R1 * [1 + α * (T2 - T1)]
R2 = (89.0 / I^2) * [1 + 0.00350 * (741 - (-37))]
R2 = (89.0 / I^2) * [1 + 0.00350 * 778]
R2 = (89.0 / I^2) * (1 + 2.723)
R2 = (89.0 / I^2) * 3.723
R2 ≈ 331.347 / I^2
Step 3:
P2 = I^2 * R2
P2 = I^2 * (331.347 / I^2)
P2 ≈ 331.347 W
So, when the temperature is 741ºC, the rate of energy dissipated by the wire will be approximately 331.347W.