I ask some help in understanding better the concept of voltage. The voltage is a difference in electric potential between two points $a$ and $b$. It is defined as $$V_{ab}=-\int^a_b\mathbf{E}\cdot d\mathbf{l}$$
However, I'm a bit confused with the use of notation:
- Is $V_{ab}$ the same as $\Delta{V}$, or rather $-\Delta{V}$? In fact, $V_{ab}$ is also written as $V_a-V_b$, while $\Delta{V}$ should be a difference between a final and an initial position.
- What do $a$ and $b$ represent? They are extrema of integration, but how do we select them in a problem, one as the starting position and the other as the arrival? What does the integration from one to the other (and not vice versa) mean?
Eventually, I would like to add another question, this one about the integrand:
- What is concretely $\text d\mathbf l$, and what is its direction?
Thanks very much!
Edit: My problem is somehow similar to the one expressed here: is the voltage defined as $\Delta{V}$ or $-\Delta{V}$? Sometimes I've found it is the work per unit charge done by the electric field, other times the work by an external force against the electric field. What is the correct definition?
Mabe I'm confusing the difference in potential and the voltage drop?