Askiitians Tutor Team
Last Activity: 4 Months ago
To tackle this problem, we need to analyze the thermal conduction between two metal cubes made of copper and aluminum. We’ll break it down into two parts: calculating the total current from one reservoir to the other and determining the ratio of the thermal currents carried by the two cubes. Let's dive into the details.
Understanding Thermal Conductivity
Thermal conductivity is a measure of how well a material can conduct heat. The formula for the heat transfer (thermal current) through a material is given by Fourier's law of heat conduction:
Q = k × A × (ΔT / d)
- Q = heat transfer per unit time (thermal current)
- k = thermal conductivity of the material (W/m·K)
- A = cross-sectional area through which heat is conducted (m²)
- ΔT = temperature difference across the material (K)
- d = thickness of the material (m)
Given Data
For both cubes, we have:
- Edge length of the cubes = 3 cm = 0.03 m
- Thermal conductivity of copper (kCu) = 401 W/m·K
- Thermal conductivity of aluminum (kAl) = 205 W/m·K (approximate value)
Calculating the Cross-Sectional Area
The cross-sectional area (A) for each cube can be calculated as:
A = side × side = 0.03 m × 0.03 m = 0.0009 m²
Determining the Total Current
Assuming we have a temperature difference (ΔT) between the two reservoirs, we can calculate the thermal current for each cube. Let's denote the temperature difference as ΔT (in Kelvin).
For the copper cube:
QCu = kCu × A × (ΔT / d)
QCu = 401 W/m·K × 0.0009 m² × (ΔT / 0.03 m)
QCu = 401 × 0.0009 × (ΔT / 0.03)
For the aluminum cube:
QAl = kAl × A × (ΔT / d)
QAl = 205 W/m·K × 0.0009 m² × (ΔT / 0.03 m)
QAl = 205 × 0.0009 × (ΔT / 0.03)
Total Current from One Reservoir to Another
The total current (thermal current) from one reservoir to the other is the sum of the currents through both cubes:
Total Current (QTotal) = QCu + QAl
Finding the Ratio of Thermal Currents
To find the ratio of the thermal currents carried by the copper cube to that of the aluminum cube, we can express it as:
Ratio = QCu / QAl
Substituting the expressions we derived earlier:
Ratio = (401 × 0.0009 × (ΔT / 0.03)) / (205 × 0.0009 × (ΔT / 0.03))
Notice that the terms (0.0009) and (ΔT / 0.03) cancel out:
Ratio = 401 / 205
Calculating this gives:
Ratio ≈ 1.95
Final Results
In summary, the total current from one reservoir to the other depends on the temperature difference and can be calculated using the formulas provided. The ratio of the thermal current carried by the copper cube to that carried by the aluminum cube is approximately 1.95, indicating that copper conducts heat nearly twice as effectively as aluminum under the same conditions.
