Section 25.2 Finding \(\Delta\)V from E
Subsubsection Key Ideas
If you know the electric field of a given charge distribution, you can integrate the electric field along a path to determine the electric potential difference between two points. Notice that the electric potential difference is a scalar quantity.
Principle 25.2.2. Determining the Electric Potential Difference from the Electric Field.
The electric potential difference can be calculated from a known electric field using the equation
\begin{equation*}
\Delta V = - \int_{r_i}^{r_f} \vec{E} \cdot d\vec{r}\text{.}
\end{equation*}
Definition 25.2.3. Cartesian \(\vec{dr}\).
In the equation for \(\Delta V\) above, the quantity \(\vec{dr}\) represents an infinitesimally small step along a path from \(r_i\) to \(r_f\text{.}\) In Cartesian coordinates, such a step may be represented as:
\begin{equation*}
\vec{dr} = dx\hat{x} + dy\hat{y} + dz\hat{z}
\end{equation*}
Given this definition, \(\vec{dr}\) will not be the same for different paths, and you will typically need to use your knowledge of the specific path to simplify \(\vec{dr}.\)
Notation 25.2.4. Shortcuts for \(\vec{dr}\).
The simplest paths can be described as along a single direction: for example, if the path is only in the \(y\)-direction, \(\vec{dr} = dy \hat{y}\text{,}\) while if the path is only in the \(z\)-direction then \(\vec{dr} = dz \hat{z}\text{.}\)
Path Independence.
The electric field is an example of a conservative vector field. One consequence of this is that the line integral for finding \(\Delta V\) is path independent. Since the value of the line integral does not depend on the path, you can always choose the path that makes evaluating the integral as easy as possible.
Subsubsection Activities
Activity 25.2.1. Sensemaking.
There is a negative sign in front of the integral in the above equation for \(\Delta V\text{.}\) What do you think this negative sign means physically?
