Question about voltage difference?

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SUMMARY

The correlation between the greatest voltage and the direction of electric field lines is established through the relationship defined by the equation for potential difference, ΔV = -qEΔx cos(θ). This equation indicates that the potential difference is maximized when the angle θ between the electric field vector and the displacement vector is zero degrees, meaning the charge moves in the direction of the electric field. The discussion emphasizes the importance of understanding uniform electric fields to simplify calculations, avoiding the complexities introduced by calculus.

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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:

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

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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