Three rods of lengths 20cm each and area of cross section `1cm^(2)` are joined to from a triangle ABC. The conductivities of the rods are `k_(AB)=50Js^(-1)m^(-1)C^(-1)` , `k_(BC)=200Js^(-1)m^(-1)C^(-1)` , and `k_(AC)=400Js^(-1)m^(-1)C^(-1)` . The junction A,B and C are maintained at `40^(@)C` , `80^(@)C` and `80^(@)C` respectively. Find the rate of heat flowing through the rods AB, AC and BC .

Explanation

To solve the problem, we need to calculate the rate of heat transfer through each of the rods AB, AC, and BC using Fourier's law of heat conduction. The formula for the rate of heat transfer (Q/t) through a rod is given by:

\[ \frac{Q}{t} = k \cdot A \cdot \frac{\Delta T}{L} \]

Where:
- \( k \) = thermal conductivity of the material (in \( \text{Js}^{-1}\text{m}^{-1}\text{C}^{-1} \))
- \( A \) = cross-sectional area (in \( \text{m}^2 \))
- \( \Delta T \) = temperature difference across the rod (in °C)
- \( L \) = length of the rod (in m)

### Step 1: Calculate the rate of heat flow through rod AB

1. **Identify the parameters for rod AB:**
- \( k_{AB} = 50 \, \text{Js}^{-1}\text{m}^{-1}\text{C}^{-1} \)
- \( A = 1 \, \text{cm}^2 = 1 \times 10^{-4} \, \text{m}^2 \)
- \( \Delta T = T_B - T_A = 80°C - 40°C = 40°C \)
- \( L = 20 \, \text{cm} = 0.2 \, \text{m} \)

2. **Substitute the values into the formula:**
\[ \frac{Q}{t}_{AB} = 50 \cdot (1 \times 10^{-4}) \cdot \frac{40}{0.2} \]

3. **Calculate:**
\[ \frac{Q}{t}_{AB} = 50 \cdot (1 \times 10^{-4}) \cdot 200 = 1 \, \text{W} \]

### Step 2: Calculate the rate of heat flow through rod AC

1. **Identify the parameters for rod AC:**
- \( k_{AC} = 400 \, \text{Js}^{-1}\text{m}^{-1}\text{C}^{-1} \)
- \( A = 1 \, \text{cm}^2 = 1 \times 10^{-4} \, \text{m}^2 \)
- \( \Delta T = T_C - T_A = 80°C - 40°C = 40°C \)
- \( L = 20 \, \text{cm} = 0.2 \, \text{m} \)

2. **Substitute the values into the formula:**
\[ \frac{Q}{t}_{AC} = 400 \cdot (1 \times 10^{-4}) \cdot \frac{40}{0.2} \]

3. **Calculate:**
\[ \frac{Q}{t}_{AC} = 400 \cdot (1 \times 10^{-4}) \cdot 200 = 8 \, \text{W} \]

### Step 3: Calculate the rate of heat flow through rod BC

1. **Identify the parameters for rod BC:**
- \( k_{BC} = 200 \, \text{Js}^{-1}\text{m}^{-1}\text{C}^{-1} \)
- \( A = 1 \, \text{cm}^2 = 1 \times 10^{-4} \, \text{m}^2 \)
- \( \Delta T = T_C - T_B = 80°C - 80°C = 0°C \)
- \( L = 20 \, \text{cm} = 0.2 \, \text{m} \)

2. **Substitute the values into the formula:**
\[ \frac{Q}{t}_{BC} = 200 \cdot (1 \times 10^{-4}) \cdot \frac{0}{0.2} \]

3. **Calculate:**
\[ \frac{Q}{t}_{BC} = 200 \cdot (1 \times 10^{-4}) \cdot 0 = 0 \, \text{W} \]

### Final Results:
- Rate of heat flow through rod AB: \( 1 \, \text{W} \)
- Rate of heat flow through rod AC: \( 8 \, \text{W} \)
- Rate of heat flow through rod BC: \( 0 \, \text{W} \)