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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 current of I1 = 1.89A running across it when the temperature of the wire is 71.0∘ . If the voltage from the “bat”tery remains constant then what would be the current flowing across the wire when the temperature is 158.0∘? The temperature coefficient of resistivity for the wire is 2.75x10^-3(∘)−1.

User Dubnde
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1 Answer

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Step-by-step explanation:

To find the current flowing across the wire when the temperature is 158.0∘, we can use the concept of temperature dependence of resistance in a conductor.

The formula to calculate the resistance (R) of a conductor with temperature coefficient of resistivity (α) is given by:

R2 = R1 * [1 + α * (T2 - T1)]

where:

R1 = Resistance of the conductor at temperature T1

R2 = Resistance of the conductor at temperature T2

α = Temperature coefficient of resistivity

T1 = Initial temperature

T2 = Final temperature

In this case, we are given that the current (I1) across the wire is 1.89A when the temperature (T1) is 71.0∘. We need to find the current (I2) when the temperature (T2) is 158.0∘.

Step 1: Find the resistance at T1:

The current (I1) is given as 1.89A, and we know that V = I * R, where V is voltage and R is resistance. Since the voltage is constant, the resistance at T1 (R1) is given by:

R1 = V / I1

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 current at T2:

The current at T2 (I2) can be calculated using Ohm's law:

I2 = V / R2

Now, let's calculate the values:

Given data:

I1 = 1.89A (current at T1)

T1 = 71.0∘ (initial temperature)

T2 = 158.0∘ (final temperature)

α = 2.75x10^-3(∘)−1 (temperature coefficient of resistivity)

Step 1:

R1 = V / I1 (Since V is constant, we can ignore it for the purpose of the calculation.)

R1 = 1 / 1.89 ≈ 0.5291 ohms

Step 2:

R2 = R1 * [1 + α * (T2 - T1)]

R2 = 0.5291 * [1 + 2.75x10^-3 * (158.0 - 71.0)]

R2 ≈ 0.5291 * [1 + 2.75x10^-3 * 87.0]

R2 ≈ 0.5291 * (1 + 0.23925)

R2 ≈ 0.5291 * 1.23925

R2 ≈ 0.6559 ohms

Step 3:

I2 = V / R2 (Since V is constant, we can ignore it for the purpose of the calculation.)

I2 ≈ 1 / 0.6559 ≈ 1.524A

So, when the temperature is 158.0∘, the current flowing across the wire will be approximately 1.524A.

User Sarath Joseph
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