Three rods of identical cross-sectional area and made from the same metal form the sides of an equilateral triangle `ABC` The points `A` and `B` are maintained at temperature `sqrt3 T` and `T` respectively In the steady state, the temperature of the point `C` is `T_(C)` Assuming that only heat conduction takes place, the value of `T_(C)//T` is equal to .

Options

A. `(1+sqrt3)/(2)`
B. `(1-sqrt3)/(2)`
C. `(1+sqrt2)/(2)`
D. `(1-sqrt2)/(2)`

Answer

To solve the problem, we need to analyze the heat conduction in the three rods forming an equilateral triangle. The rods are denoted as AB, BC, and CA, and we know the temperatures at points A and B, as well as the temperature at point C (which we will denote as \( T_C \)).

### Step-by-Step Solution:

1. **Identify the Given Temperatures:**
- Temperature at point A: \( T_A = \sqrt{3}T \)
- Temperature at point B: \( T_B = T \)
- Temperature at point C: \( T_C \) (unknown)

2. **Understand the Heat Conduction:**
- In a steady state, the rate of heat conduction through each rod must be equal. The heat conduction can be expressed as:
\[ \frac{T_A - T_C}{R_{CA}} = \frac{T_C - T_B}{R_{BC}} \]
- Here, \( R_{CA} \) and \( R_{BC} \) are the thermal resistances of rods CA and BC, respectively.

3. **Calculate the Thermal Resistances:**
- The thermal resistance \( R \) for a rod can be expressed as:
\[ R = \frac{L}{kA} \]
- Since all rods have the same length \( L \), area \( A \), and are made of the same material (same thermal conductivity \( k \)), we have:
\[ R_{CA} = R_{BC} = \frac{L}{kA} \]

4. **Set Up the Equation:**
- Using the thermal resistances, we can rewrite the heat conduction equation:
\[ \frac{\sqrt{3}T - T_C}{\frac{L}{kA}} = \frac{T_C - T}{\frac{L}{kA}} \]
- The \( \frac{L}{kA} \) cancels out from both sides:
\[ \sqrt{3}T - T_C = T_C - T \]

5. **Rearranging the Equation:**
- Rearranging gives:
\[ \sqrt{3}T - T = 2T_C \]
- This simplifies to:
\[ T_C = \frac{\sqrt{3}T - T}{2} = \frac{(\sqrt{3} - 1)T}{2} \]

6. **Finding the Ratio \( \frac{T_C}{T} \):**
- Now, we can express the ratio:
\[ \frac{T_C}{T} = \frac{\frac{(\sqrt{3} - 1)T}{2}}{T} = \frac{\sqrt{3} - 1}{2} \]

7. **Final Result:**
- Therefore, the value of \( \frac{T_C}{T} \) is:
\[ \frac{T_C}{T} = \frac{\sqrt{3} - 1}{2} \]