Four identical rods of same material are joined end to end to form a square. If the temperature difference between the ends of a diagonal is `100^(@)C`, then the temperature difference between the ends of other diagonal will be

Options

A. `0^(@)C`
B. `(100)/(l)""^(@)C`
C. `(100)/(2l)""^(@)C`
D. `100^(@)C`

Explanation

To solve the problem, we need to analyze the thermal properties of the square formed by the four identical rods. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Configuration
We have four identical rods of the same material joined end to end to form a square. Let's label the corners of the square as A, B, C, and D.

### Step 2: Identify the Temperature Difference
According to the problem, the temperature difference between the ends of one diagonal (from A to C) is given as 100°C. This means:
\[ \theta_A - \theta_C = 100°C \]

### Step 3: Analyze the Other Diagonal
We need to find the temperature difference between the ends of the other diagonal (from B to D). We denote this temperature difference as:
\[ \theta_B - \theta_D \]

### Step 4: Consider Thermal Resistance
Since all rods are identical, they have the same length, cross-sectional area, and thermal conductivity. Therefore, the thermal resistance (R) for each rod is the same.

### Step 5: Apply Heat Current Concept
The heat current (S) flowing through the rods can be expressed as:
\[ S = \frac{\Delta T}{R} \]
where \(\Delta T\) is the temperature difference across the rod and R is the thermal resistance.

### Step 6: Relate Heat Currents
Since the thermal resistance is the same for both diagonals, the heat current through diagonal AC (from A to C) and diagonal BD (from B to D) must also be equal. Thus:
\[ S_{AC} = S_{BD} \]

### Step 7: Set Up the Equations
For diagonal AC:
\[ S_{AC} = \frac{\theta_A - \theta_C}{R} \]
For diagonal BD:
\[ S_{BD} = \frac{\theta_B - \theta_D}{R} \]

Since \(S_{AC} = S_{BD}\), we can equate the two equations:
\[ \frac{\theta_A - \theta_C}{R} = \frac{\theta_B - \theta_D}{R} \]
This simplifies to:
\[ \theta_A - \theta_C = \theta_B - \theta_D \]

### Step 8: Substitute Known Values
We know that \(\theta_A - \theta_C = 100°C\), so we can substitute this into the equation:
\[ 100°C = \theta_B - \theta_D \]

### Step 9: Conclusion
Since the heat current through both diagonals is equal and the rods are identical, the temperature difference between the ends of diagonal BD must be zero:
\[ \theta_B - \theta_D = 0 \]

Thus, the temperature difference between the ends of the other diagonal (BD) is:
\[ \theta_B - \theta_D = 0°C \]

### Final Answer
The temperature difference between the ends of the other diagonal is **0°C**.

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