Three rods `A,B` and `C` have the same dimensions Their conductivities are `K_(A)` K and `K_(C)` respectively `A` and `B` are placed end to end with their free ends kept at certain temperature difference `C` is placed separately with its ends kept at same temperature difference The two arrangements conduct heat at the same rate `K_(c)` must be equal to .

Options

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

Explanation

To solve the problem, we need to analyze the heat conduction through the rods A, B, and C. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Configuration
- Rods A and B are placed end to end, and their free ends are kept at a temperature difference (ΔT).
- Rod C is placed separately with its ends also kept at the same temperature difference (ΔT).
- All rods have the same dimensions (length L and cross-sectional area A).

**Hint:** Visualize the arrangement of the rods and the temperature differences applied to them.

### Step 2: Determine 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 rod A:
\[ R_A = \frac{L}{K_A \cdot A} \]
- For rod B:
\[ R_B = \frac{L}{K_B \cdot A} \]
- For rod C:
\[ R_C = \frac{L}{K_C \cdot A} \]

**Hint:** Remember that thermal resistance is inversely proportional to thermal conductivity.

### Step 3: Calculate the Total Resistance for A and B in Series
- Since rods A and B are in series, the total thermal resistance (R_total) of A and B is:
\[ R_{total} = R_A + R_B = \frac{L}{K_A \cdot A} + \frac{L}{K_B \cdot A} \]
- This can be simplified to:
\[ R_{total} = \frac{L}{A} \left( \frac{1}{K_A} + \frac{1}{K_B} \right) \]

**Hint:** In series, resistances add up.

### Step 4: Set Up the Heat Flow Equations
- The heat flow rate (Q) through rods A and B can be expressed as:
\[ Q = \frac{\Delta T}{R_{total}} = \frac{\Delta T}{\frac{L}{A} \left( \frac{1}{K_A} + \frac{1}{K_B} \right)} \]
- The heat flow rate through rod C is:
\[ Q = \frac{\Delta T}{R_C} = \frac{\Delta T}{\frac{L}{K_C \cdot A}} \]

**Hint:** Heat flow rate is directly proportional to the temperature difference and inversely proportional to the resistance.

### Step 5: Equate the Heat Flow Rates
- Since both arrangements conduct heat at the same rate, we can set the two equations for Q equal to each other:
\[ \frac{\Delta T}{\frac{L}{A} \left( \frac{1}{K_A} + \frac{1}{K_B} \right)} = \frac{\Delta T}{\frac{L}{K_C \cdot A}} \]

**Hint:** The temperature difference (ΔT) and length (L) can be canceled out from both sides.

### Step 6: Solve for K_C
- After canceling out the common terms, we get:
\[ \frac{1}{K_C} = \frac{1}{K_A} + \frac{1}{K_B} \]
- Taking the reciprocal gives us:
\[ K_C = \frac{K_A K_B}{K_A + K_B} \]

**Hint:** This is a form of the parallel resistance formula, but applied to thermal conductivities.

### Final Answer
Thus, the thermal conductivity \(K_C\) must be equal to:
\[ K_C = \frac{K_A K_B}{K_A + K_B} \]