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.