Potential DIfference Path Independence, claimed by Aditya Mohile

Primary Concept

The voltage between 2 points does not depend on the path taken to travel from one point to another.

A Mathematical Understanding

[math]\displaystyle{ \Delta V_{AB}= - \int^B_A \vec{E} \cdot d\vec{l} }[/math]

The potential difference from A to B is found by integrating the electric field with respect to the distance between the two points.

[math]\displaystyle{ \Delta V_{AB}= {E} \int^B_A d\vec{l} }[/math]

Here, E is uniform and constant and can be isolated outside of the integral.

[math]\displaystyle{ \Delta V_{AB} = {E}\Delta l_{AB} }[/math]

From this final statement, The potential difference is equivalent to the uniform electric field in the region multiplied by the change in position of the charge from A to B. As a result, it is clear that regardless of the how we get from A to B the only distance that matters is the distance between the final and initial point of the charge acted on. This indicates that the potential difference is "path independent".

A Visual Representation