Three identical rods A, B and C are placed end to end. A temperature difference is maintained between the free ends of A and C. The thermal conductitvity of B is thrice that of C and half that of A. The effective thermal conductivity of rod A)

Options

A. `(1)/(3)K_(A)`
B. `3 K_(A)`
C. `2 K_(A)`
D. `(2)/(3)K_(A)`

Explanation

To find the effective thermal conductivity of the three rods A, B, and C placed end to end, we can follow these steps:

### Step 1: Define the thermal conductivities
Let:
- \( K_C \) = thermal conductivity of rod C
- \( K_B = 3K_C \) = thermal conductivity of rod B (given that it is thrice that of C)
- \( K_A = 2K_B \) = thermal conductivity of rod A (given that it is half that of A)

From the above relationships, we can express \( K_A \) in terms of \( K_C \):
- \( K_B = 3K_C \)
- \( K_A = 2K_B = 2(3K_C) = 6K_C \)

### Step 2: Calculate the thermal resistance of each rod
The thermal resistance \( R \) of a rod can be calculated using the formula:
\[ R = \frac{L}{K \cdot A} \]
where \( L \) is the length of the rod, \( K \) is the thermal conductivity, and \( A \) is the cross-sectional area.

For the three rods:
- \( R_A = \frac{L}{K_A \cdot A} = \frac{L}{6K_C \cdot A} \)
- \( R_B = \frac{L}{K_B \cdot A} = \frac{L}{3K_C \cdot A} \)
- \( R_C = \frac{L}{K_C \cdot A} = \frac{L}{K_C \cdot A} \)

### Step 3: Calculate the total thermal resistance
Since the rods are in series, the total thermal resistance \( R_{total} \) is the sum of the individual resistances:
\[ R_{total} = R_A + R_B + R_C \]
Substituting the values:
\[ R_{total} = \frac{L}{6K_C \cdot A} + \frac{L}{3K_C \cdot A} + \frac{L}{K_C \cdot A} \]

### Step 4: Simplify the expression
To simplify, we can find a common denominator, which is \( 6K_C \cdot A \):
\[ R_{total} = \frac{L}{6K_C \cdot A} + \frac{2L}{6K_C \cdot A} + \frac{6L}{6K_C \cdot A} \]
\[ R_{total} = \frac{L + 2L + 6L}{6K_C \cdot A} = \frac{9L}{6K_C \cdot A} = \frac{3L}{2K_C \cdot A} \]

### Step 5: Calculate the effective thermal conductivity
The effective thermal conductivity \( K_{eff} \) can be calculated using the formula:
\[ R_{total} = \frac{L}{K_{eff} \cdot A} \]
Equating the two expressions for \( R_{total} \):
\[ \frac{3L}{2K_C \cdot A} = \frac{L}{K_{eff} \cdot A} \]
Cancelling \( L \) and \( A \) from both sides:
\[ \frac{3}{2K_C} = \frac{1}{K_{eff}} \]
Thus, rearranging gives:
\[ K_{eff} = \frac{2K_C}{3} \]

### Final Answer
The effective thermal conductivity of the three rods is:
\[ K_{eff} = \frac{2K_C}{3} \]