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The decrease in potential across the wir...

The decrease in potential across the wire is _______ proportional to the length of the wire.

A

Directly

B

Inversely

C

Both (a) and (b)

D

None

Text Solution

AI Generated Solution

The correct Answer is:
To solve the question regarding the relationship between the decrease in potential across a wire and its length, we can break it down into the following steps: ### Step-by-Step Solution: 1. **Understanding the Concept of Potential Difference**: - The potential difference (V) across a wire is the difference in electric potential between two points in the wire. It is caused by the flow of electric current through the wire. 2. **Assuming a Uniform Wire**: - We consider a uniform wire, which means that the cross-sectional area (A) of the wire is constant along its length (L). This uniformity allows us to analyze the relationship between potential difference and length more straightforwardly. 3. **Current Flow and Electric Field**: - When a current (I) flows through the wire, it creates an electric field (E) within the wire. The electric field is directed from the higher potential to the lower potential. 4. **Relationship Between Electric Field and Potential**: - The relationship between electric field (E) and potential difference (V) over a length (L) of wire can be expressed as: \[ E = -\frac{dV}{dr} \] - Integrating this equation over the length of the wire gives us: \[ V = -E \cdot L \] - This indicates that the potential difference is directly related to the length of the wire. 5. **Proportionality of Potential Decrease to Length**: - Since the potential decreases as you move along the wire in the direction of the electric field, we can conclude that the decrease in potential (ΔV) across the wire is directly proportional to the length (L) of the wire: \[ \Delta V \propto L \] - This means that as the length of the wire increases, the potential difference (or decrease in potential) across it also increases. 6. **Conclusion**: - Therefore, the decrease in potential across the wire is **directly proportional** to the length of the wire. ### Final Answer: The decrease in potential across the wire is **directly proportional** to the length of the wire.
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