Line Integration: Electric Field, Potential Energy & Displacement Vector

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Line integration involves integrating a field along a specific path, focusing on the field's characteristics rather than the object's movement. When integrating the electric field to determine electric potential and potential energy, the displacement vector ds is treated as a scalar product with the electric field due to the nature of potential functions. This leads to the relationship E = -grad(phi), where phi represents the potential function. Understanding the gradient and calculus principles helps clarify the line integral that defines phi. This discussion enhances comprehension of the relationship between electric fields and potential energy through line integration.
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I'm not exactly sure of this definition, but this seems to be it:
Line integration is the integration of a field...meaning the path that the field takes shape of and not the path of the object moving in the field. In that sense, the outcome of this integration only depends on the position the particle has at various points in the field.

Why is it that when integrating the Electric field in terms of electric potential and potential energy, that the displacement vector ds is taken to be a Scalar product with the Electric Field?
 
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Well that's from the very definition of a potential function. You're looking for a scalar function phi such that E=-grad(phi)

Then from the definition of the gradient and knowing calculus you can fish out that line integral that defines phi generally
 
oh wait...thanks for reminding me. I'm beginning to understand how it works.
 

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