PX157 - B7a - electric potential energy

introduction

the work done by a force, $\vec{F}$ , moving a vector distance, $d \vec{l}$

E = K + U

in a conservative system:

\begin{aligned} Δ E & = 0 \\ Δ K & = - Δ U \\ W_{a \to b} & = - Δ U \end{aligned}

from the definition of work:

W = \vec{F} \cdot d \vec{l} = F d l \cos θ

for work along a non-straight path:
$\begin{align*} dW &= \vec F \cdot d\vec l \\ W_{a\to b} &= \int_{a}^{b} \vec F \cdot d\vec l$

line integrals

closed path

W = \oint \vec{F} \cdot d \vec{l}

path is in the x-direction

\begin{aligned} \vec{r} & = x \hat{x} \\ d \vec{l} & = d \vec{r} = d x \hat{x} \\ \int_{a}^{b} \vec{F} \cdot d \vec{l} & = \int_{x_{a}}^{x_{b}} \vec{F} \cdot \hat{x} d x \\ = \int_{x_{a}}^{x_{b}} F_{x} d x \end{aligned}

path is in the radial direction

\begin{aligned} \vec{r} & = r \hat{r} \\ d \vec{l} & = d \vec{r} \\ = d r \hat{r} \end{aligned}

path is in the azimuthal direction

ie: path is a circle in the $x - y$ plane with a radius, $a$
$\vec{r} = a \hat{R} (ϕ)$
$d \vec{l} = d \vec{r} = a \frac{d \hat{R}}{d ϕ} d ϕ$
$\hat{R} = \cos ϕ \hat{x} + \sin ϕ \hat{y}$
$\frac{d \hat{R}}{d ϕ} = - \sin ϕ \hat{x} + \cos ϕ \hat{y} \equiv \hat{ϕ}$
$d \vec{l} = a \hat{ϕ} d ϕ$

W_{a \to b} = \int_{a}^{b} \vec{F} \cdot d \vec{l} = U_{a} - U_{b}