Question about voltage difference?

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The discussion centers on the relationship between voltage and electric field direction, emphasizing that the greatest voltage correlates with the direction of electric field lines. It highlights that the potential difference is defined as the negative work done by electric force on a charge, expressed mathematically as ΔV = -qEΔx cos θ. The conversation simplifies the scenario to a uniform electric field to avoid calculus complexities. The angle θ, which maximizes the potential difference, is also a key point of inquiry. Understanding this relationship is crucial for grasping electric field behavior and potential differences.
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Why do the greatest voltage and the direction of electric field lines correlate with one another? Is there a mathematical proof for this?
 
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To keep things simple, suppose the electric field is uniform, otherwise we have to use calculus. The potential difference between two points is defined as the negative of the work done by the electric force on a charge that moves between those points:

\Delta V = -W_e = -\vec F_e \cdot \vec{\Delta x} = -q \vec E \cdot \vec{\Delta x}= - qE \Delta x \cos \theta

where theta is the angle between the electric field and the displacement of the charge. Which angle makes the potential difference a maximum?
 

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