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The two ends of a uniform conductor are ...

The two ends of a uniform conductor are joined to a cell of e.m.f. `E` and some internal resistance. Starting from the midpoint `P` of the conductor, we move in the direction of current and return to `P`. The potential `V` at every point on the path is plotted against the distance covered `(x)`. which of the following graphs best represent the resulting curve ?

A

B

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D

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The correct Answer is:
To solve the problem, we need to analyze the behavior of potential \( V \) as we move along a uniform conductor connected to a cell with an electromotive force (e.m.f.) \( E \) and some internal resistance. ### Step-by-Step Solution: 1. **Understanding the Setup**: - We have a uniform conductor connected to a cell with e.m.f. \( E \). - The conductor has some internal resistance, which affects the potential difference across it. 2. **Identifying the Midpoint**: - We start from the midpoint \( P \) of the conductor. This is our reference point where we will measure the potential. 3. **Moving in the Direction of Current**: - As we move in the direction of the current from point \( P \) to one end of the conductor, the potential \( V \) decreases linearly. - This decrease is due to the resistance of the conductor; the potential drop across the conductor is proportional to the distance moved. 4. **Reaching the End of the Conductor**: - At the end of the conductor, the potential will be at its minimum value. Let’s denote this point as \( B \). 5. **Entering the Cell**: - When we reach the end of the conductor and start to enter the cell, the potential will increase as we move from the negative terminal to the positive terminal of the cell. - This increase is equal to the e.m.f. \( E \) of the cell, but due to the internal resistance, the potential will always be less than \( E \). 6. **Returning to the Midpoint**: - As we move back towards the midpoint \( P \), the potential will again decrease linearly as we move back through the conductor. 7. **Graph Representation**: - The graph of potential \( V \) against distance \( x \) will show a linear decrease as we move away from \( P \) to \( B \), followed by a linear increase as we move through the cell and return to \( P \). - This process will repeat, showing a sawtooth pattern in the graph. 8. **Choosing the Correct Graph**: - Among the given options, we need to select the graph that shows a linear decrease followed by a linear increase, while also ensuring that the potential remains less than \( E \) throughout the process. ### Conclusion: - The correct graph will show a linear decrease in potential as we move away from the midpoint, followed by a linear increase as we return through the cell, and this pattern will repeat.
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