AP Physics 2 Exam Question: Heat Transfer

A copper rod with a length of 0.5 m and a cross-sectional area of 0.01 m² is initially at a temperature of 300 K. One end of the rod is in contact with a heat source at a temperature of 500 K, while the other end is in contact with a heat sink at a temperature of 200 K. The thermal conductivity of copper is 400 W/(m·K) and its specific heat capacity is 390 J/(kg·K). Assume steady-state conditions.

1. Calculate the rate of heat transfer through the copper rod.
2. Determine the temperature at a point halfway along the copper rod.

Answer with Step-by-Step Explanation:

1. To find the rate of heat transfer through the copper rod, we can use Fourier's law of heat conduction:

   

   Given:

   • Length of the rod (L) = 0.5 m
   • Cross-sectional area (A) = 0.01 m²
   • Thermal conductivity of copper (k) = 400 W/(m·K)
   • Temperature difference across the rod (ΔT) = 500 K - 200 K = 300 K

   The rate of heat transfer (P) can be calculated as:

   P = (k * A * ΔT) / L

   Plugging in the given values:

   P = (400 W/(m·K) * 0.01 m² * 300 K) / 0.5 m

   P = 2400 W

   Therefore, the rate of heat transfer through the copper rod is 2400 W.

2. To determine the temperature at a point halfway along the copper rod, we can apply the concept of thermal equilibrium through conduction:

   The rate of heat transfer at any point along the rod (x) will be given by:

   P(x) = (k * A * ΔT) / L

   Since we know the rate of heat transfer through the entire rod is 2400 W, we can set up the following equation:

   P(x) = 2400 W

   (k * A * ΔT) / L = 2400 W

   Rearranging and solving for ΔT:

   ΔT = (2400 W * L) / (k * A)

   Plugging in the given values:

   ΔT = (2400 W * 0.5 m) / (400 W/(m·K) * 0.01 m²)

   ΔT = 30 K

   Therefore, the temperature at a point halfway along the copper rod is 300 K + 30 K = 330 K.