Two rods of copper and brass having the same length and cross - section are joined end to end. The free end of the copper is at `0^(@)C` and of brass is at `80^(@)C` in steady-state. If thermal conductivity of copper is 4 times of that of brass find the temp. at junction of two rods

Options

A. `80^(@)C`
B. `64^(@)C`
C. `20^(@)C`
D. `16^(@)C`

Explanation

To solve the problem, we need to find the temperature at the junction of two rods made of copper and brass. We will use the concept of thermal conductivity and the steady-state condition where the heat flow through both rods is equal.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Let the thermal conductivity of brass be \( K_B \).
- Then, the thermal conductivity of copper will be \( K_C = 4K_B \).
- Temperature at the free end of copper rod, \( T_C = 0^\circ C \).
- Temperature at the free end of brass rod, \( T_B = 80^\circ C \).
- Let the temperature at the junction be \( T_0 \).

2. **Set Up Heat Current Equations:**
- The heat current through the copper rod can be expressed as:
\[ Q_C = \frac{K_C \cdot A \cdot (T_0 - T_C)}{L} \]
- The heat current through the brass rod can be expressed as:
\[ Q_B = \frac{K_B \cdot A \cdot (T_B - T_0)}{L} \]
- Here, \( A \) is the cross-sectional area, and \( L \) is the length of the rods.

3. **Equate the Heat Currents:**
- Since we are in steady state, the heat current through both rods must be equal:
\[ Q_C = Q_B \]
- Substituting the expressions for \( Q_C \) and \( Q_B \):
\[ \frac{K_C \cdot A \cdot (T_0 - 0)}{L} = \frac{K_B \cdot A \cdot (80 - T_0)}{L} \]
- We can cancel \( A \) and \( L \) from both sides:
\[ K_C \cdot T_0 = K_B \cdot (80 - T_0) \]

4. **Substitute \( K_C = 4K_B \):**
- Replacing \( K_C \) in the equation:
\[ 4K_B \cdot T_0 = K_B \cdot (80 - T_0) \]
- Dividing both sides by \( K_B \) (assuming \( K_B \neq 0 \)):
\[ 4T_0 = 80 - T_0 \]

5. **Solve for \( T_0 \):**
- Rearranging the equation gives:
\[ 4T_0 + T_0 = 80 \]
\[ 5T_0 = 80 \]
\[ T_0 = \frac{80}{5} = 16^\circ C \]

6. **Conclusion:**
- The temperature at the junction of the two rods is \( T_0 = 16^\circ C \).