- #1

physics girl phd

- 931

- 3

## Homework Statement

I'm trying to do a problem two ways, and things aren't consistent, finding the electric potential difference in a linearly varying field.

The electric potential difference between two points is often summarized in texts as ΔV = Vf - Vi = - ∫ E⋅ds where the lower bound of integration is the initial point and the upper bound of integration is the final. It's my understanding the vector ds points from the initial point to the final point.

My example is a field with linear variation, pointing in the +i direction, with value E = -Bx , and the points we are concerned with are x1 and x2, which are both in the +x region. I would expect to be able to independently find V2-V1, and V1-V2, and find they are equal and opposite. But for some reason I'm not getting that. When the bounds of integration flip, and the path vector ds from one point to another flips, you get two negatives that cancel the effects, so you get the same ΔV, not an equal but opposite ΔV.

I attach my work.

From what I can tell looking at various texts I have (from introductory to intermediate... I haven't yet pulled out grad-level), their definitions of ds are often vague, and they don't change its direction in examples... can someone clarify this?

## Homework Equations

ΔV = Vf - Vi = - ∫ E⋅ds where the lower bound of integration is the initial point (or reference) and the upper bound of integration is the final. It's my understanding the vector ds points from the initial point to the final point (but I could be wrong there).

## The Attempt at a Solution

I attach my work. I get the correct delta V one way... but don't get the equal and opposite the other (which is from the further point inward.