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Bruce Wayne is working on a new device that is going to be powered by one of his patented “bat”teries. Inside of the device is a wire that has a dissipates energy at a rate of P1 = 89.0W running across it when the temperature of the wire is −37.0ºC. If the current from the “bat”tery remains constant then what would be the rate of energy dissipated by the wire when the temperature is 741ºC ? The temperature coefficient of resistivity for the wire is 0.00350(∘)−1.

1 Answer

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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.

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