Hi. I'm wondering how one determines the direction of dl (vector) when calculating V, the electric potential. So, suppose we want to calculate the electric potential caused by a point charge q at a distance a from the point charge, if we know the electric field of a point charge.
The textbook I have says to calculate the potential by Va - Vb = integral from a to b of E*dl.
The Attempt at a Solution
So, I can let b = infinity, to get Va - Vinfinity = integral from a to infinity of k*q/r2 dr, where I have said that dl = dr*rhat, since I am integrating from r = a to r = infinity. Doing the integral, and assuming Vinfinity = 0, I get the correct answer,
V = kq/a.
But suppose instead that I had integrated from infinity to a instead of the other way. So,
Vinfinity - Va = integral from infinity to a of E*dl.
Now, since I am coming from infinity toward a, I make dl = -dr*rhat. That introduces one negative sign. But the order of integration is also flipped, which introduces another. BUT, since I have Vinfinity - Va, that introduces a third negative, so I end up with
V = -kq/a
the opposite of what I got before, and the wrong answer. Where did I go wrong? I'm guessing it must have been when I said that dl = -dr*rhat - but why isn't that correct? What sets the proper direction of dl?