Question:

A metallic bar of length 2.0 m and uniform cross-sectional area of 0.04 m^2 is initially at a temperature of 20 °C. The bar is heated to a final temperature of 80 °C by passing a current of 10 A through it for 5 minutes. The specific heat capacity of the metal is 450 J/kg·°C, and its resistivity is 4.0 × 10^(-7) Ω·m.

a) Determine the total heat transferred to the metal bar during the 5 minutes. b) Calculate the change in resistivity of the bar due to the temperature increase. c) Find the power dissipated in the bar during the heating process.

Answer:

a) The total heat transferred to the metal bar during the 5 minutes can be calculated using the formula:

Heat transferred = (mass) × (specific heat capacity) × (change in temperature)

First, we need to calculate the mass of the metal bar. The volume of the bar can be calculated using its cross-sectional area and length:

Volume = (cross-sectional area) × (length)

Next, we can find the mass using the equation:

mass = (density) × (volume)

Assuming the density of the metal is constant, we can substitute the values into the equation to find the mass:

density = mass / volume

Now we can substitute the values into the equation for heat transferred:

Heat transferred = (mass) × (specific heat capacity) × (change in temperature)

Given: Length (L) = 2.0 m Cross-sectional area (A) = 0.04 m^2 Initial temperature (T1) = 20 °C Final temperature (T2) = 80 °C Specific heat capacity (c) = 450 J/kg·°C

Density of the metal bar: Assume that the cross-sectional area is constant throughout the length of the bar. The density can be calculated as the mass divided by the volume:

mass = density × volume

Using the given values, we can find the volume and mass:

Volume = (cross-sectional area) × (length) = (0.04 m^2) × (2.0 m) = 0.08 m^3

Assuming the density of the metal bar is uniform, we can calculate it using the equation:

density = mass / volume

density = 0.02 g/cm^3 = 2000 kg/m^3

mass = 2000 kg/m^3 × 0.08 m^3 = 160 kg

Now we can substitute the values into the equation for heat transferred:

Heat transferred = (mass) × (specific heat capacity) × (change in temperature)

Heat transferred = (160 kg) × (450 J/kg·°C) × (80 °C - 20 °C)

Heat transferred = 5,760,000 J

Therefore, the total heat transferred to the metal bar during the 5 minutes is 5,760,000 J.

b) The change in resistivity (Δρ) of the bar due to the temperature increase can be calculated using the formula:

Δρ = ρ₀ × α × ΔT

where: ρ₀ is the initial resistivity of the bar α is the temperature coefficient of resistivity ΔT is the change in temperature

Given: Initial resistivity (ρ₀) = 4.0 × 10^(-7) Ω·m Change in temperature (ΔT) = (Final temperature) - (Initial temperature) = 80 °C - 20 °C = 60 °C Temperature coefficient of resistivity (α) = Assume the temperature coefficient of resistivity is constant

Substituting the values into the formula:

Δρ = (4.0 × 10^(-7) Ω·m) × (α) × (60 °C)

We are not given the value of α, so we cannot determine the exact change in resistivity without further information.

c) The power dissipated in the bar during the heating process can be calculated using Ohm's Law:

Power dissipated = (current)^2 × (resistance)

First, we need to determine the resistance of the bar. The resistance (R) can be calculated using the formula:

resistance = (resistivity) × (length / cross-sectional area)

Given: Length (L) = 2.0 m Cross-sectional area (A) = 0.04 m^2 Resistivity (ρ) = 4.0 × 10^(-7) Ω·m

Substituting the values into the formula:

resistance = (4.0 × 10^(-7) Ω·m) × (2.0 m / 0.04 m^2)

resistance = 2 Ω

Now we can substitute the values into the formula for power dissipated:

Power dissipated = (current)^2 × (resistance)

Power dissipated = (10 A)^2 × (2 Ω)

Power dissipated = 200 W

Therefore, the power dissipated in the bar during the heating process is 200 W.