The ends of a metre stick are maintain at `100^(@)C` and `0^(@)C` . One end of a rod is maintained `25^(@)C` . Where should its other end be touched on the metre stick so that there is no heat current in the rod in steady state?

Explanation

To solve the problem, we need to determine where to touch the other end of the rod on the meter stick so that there is no heat current in the rod in a steady state.

### Step-by-step Solution:

1. **Understanding the Setup**:
- One end of the meter stick is at \(100^\circ C\) and the other end is at \(0^\circ C\).
- One end of the rod is maintained at \(25^\circ C\).
- We need to find the position on the meter stick where the other end of the rod should be placed to ensure that there is no heat flow through the rod.

2. **Temperature Distribution**:
- The temperature along the meter stick varies linearly from \(0^\circ C\) to \(100^\circ C\) over a length of \(100\) cm.
- Therefore, the temperature \(T\) at any point \(x\) cm from the colder end (0 cm) can be expressed as:
\[ T(x) = \frac{100}{100} \cdot x = x \quad \text{(in degrees Celsius)} \]

3. **Condition for No Heat Current**:
- For there to be no heat current in the rod, the temperature at the point where the rod touches the meter stick must also be \(25^\circ C\).
- We need to find the value of \(x\) such that \(T(x) = 25^\circ C\).

4. **Setting Up the Equation**:
- From our temperature distribution equation:
\[ T(x) = x = 25 \]

5. **Solving for \(x\)**:
- Thus, we find:
\[ x = 25 \text{ cm} \]

6. **Conclusion**:
- The other end of the rod should be touched at \(25\) cm from the colder end of the meter stick.

### Final Answer:
The other end of the rod should be touched at \(25\) cm from the colder end of the meter stick.

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