Three rods of same material and of equal cross sectional area and length have been connected together as shown in the figure. The temperature of the junction of the rods approximately is

Options

A. `35^@C`
B. `47^@C`
C. `51^@C`
D. `57^@C`

Explanation

To find the temperature at the junction of the three rods, we can follow these steps:

### Step 1: Understand the setup
We have three rods made of the same material, with equal cross-sectional areas and lengths. The rods are connected in such a way that they form a junction. The temperatures at the ends of the rods are given as \( T_1 = 70^\circ C \) and \( T_2 = 0^\circ C \).

### Step 2: Identify thermal resistance
The thermal resistance \( R \) for each rod can be expressed as:
\[ R = \frac{L}{kA} \]
where \( L \) is the length of the rod, \( k \) is the thermal conductivity of the material, and \( A \) is the cross-sectional area. Since all rods have the same length, area, and material, the thermal resistance for each rod is the same, denoted as \( R \).

### Step 3: Apply the junction law
According to the junction law, the heat flow through each rod must be equal at the junction. We can express this as:
\[ \frac{T - T_1}{R} + \frac{T - T_2}{R} + \frac{T - T_3}{R} = 0 \]
where \( T \) is the temperature at the junction, \( T_1 \) is the temperature of the first rod, \( T_2 \) is the temperature of the second rod, and \( T_3 \) is the temperature of the third rod.

### Step 4: Substitute known values
Assuming the third rod is at \( T_3 = 0^\circ C \) (as per the problem statement), we can substitute the values:
\[ \frac{T - 70}{R} + \frac{T - 0}{R} + \frac{T - 0}{R} = 0 \]

### Step 5: Simplify the equation
This simplifies to:
\[ \frac{T - 70}{R} + \frac{T}{R} + \frac{T}{R} = 0 \]
Multiplying through by \( R \) (since \( R \neq 0 \)):
\[ T - 70 + T + T = 0 \]
This leads to:
\[ 3T - 70 = 0 \]

### Step 6: Solve for \( T \)
Rearranging gives:
\[ 3T = 70 \]
\[ T = \frac{70}{3} \approx 23.33^\circ C \]

### Step 7: Conclusion
Thus, the temperature at the junction of the rods is approximately:
\[ T \approx 23.33^\circ C \]