>[!summary] Electric potential energy refers to the amount of work required to bring a charge to some point, always in reference to another point. > **Key equations:** > Electric potential energy general equation: $\Delta U = -q\int_{a}^{b}{\vec{E} \cdot \vec{ds}}$ > Electric potential energy of two point charges: $U=\frac{kqq_0}{r}$ >[!info]+ Read Time **⏱ 2 mins** # Definition Electric potential energy is the energy stored in a system of charges in relation to their position. It's the [[Work|work]] required to bring a [[Charge|charge]] from some reference point (usually infinity) to a specific point against the [[Electric Fields|electric field]] Mathematically, we define it as the work done on a charge in an electric field over a displacement. With units [[Coulomb|coulomb]] $\Delta U = -q\int_{a}^{b}{\vec{E} \cdot \vec{ds}}$ ![[elpe_1.png|400]] [^1] >[!note] Explanation Example of a point charge in an electric field. > The path against the electric field is high potential energy since it would **require a lot of energy** to move that charge against the field. > The path towards the electric field is low energy since it would require a **less energy** to move that charge there. # Deriving Electric Potential Energy > [!warning] Assumptions In order to derive the equation for electric potential energy, assume the following: > - From the definition, the electric potential energy is the work over displacement r $W = \int _a^b \vec{F}\cdot \vec{ds}$ > - The force is the electric field $\vec{F} = q\vec{E}$ $ \begin{array}{c} \\ W = \int_{a}^b \vec{F} \cdot \vec{ds} \\ \\ \vec{F} = q\vec{E} \\ \\ W = q\int_{a}^b \vec{E}\cdot \vec{ds} \\ \\ \text{Note that $W = -\Delta U$ so now:}\\ \Delta U = -q \int_{a}^b \vec{E} \cdot \vec{ds} \end{array} $ # Deriving Electric Potential of Point Charge >[!warning] Assumptions To derive the electric potential energy of a point charge. Assume the following: > - The charge is positive >- The [[Electric Force|electric force]] is $\vec{F} = q\vec{E}$ >- From [[Coulomb's Law|Coulomb's law]] the electric field for a point charge is $E = \frac{q}{4\pi r^2 \epsilon _0}$ >- [[Work|Work]] is $W = \int_r ^{r_f} \vec{F(r)} \cdot \vec{dx}$ >- The [[Electric Fields|electric field]] is constant ![[elpe_2.png]] >[!note] Explanation Work done the electric field is independent of the path it takes (gray dotted lines) With the force being with the direction of the field $\begin{array}{c} W = \int_r ^{r_f} F(r) \cdot ds \\ W = \int_r ^{r_f}qE \cdot ds \\ \\ \text{Remeber $W = -\Delta U$} \\ \Delta U = -\int_r ^{r_f}qE \cdot ds \\ \Delta U = -q_1Er \\ \Delta U = -q_1 ( \frac{q_2}{4\pi r^2 \epsilon _0}) (r) \\ \Delta U = \frac{-q_1 q_2}{4\pi r \epsilon _0} \end{array}$ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/j3GrOKre__0?si=ktndbPOJ4wfQg4ZA" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> <iframe width="560" height="315" src="https://www.youtube.com/embed/QpVxj3XrLgk?si=E7XATZxY9AeOXBPr" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> --- > 💡 Found this concept helpful? [Star Math & Matter on GitHub](https://github.com/rajeevphysics/Obsidan-MathMatter) to support more intuitive science breakdowns like this. --- [^1]: Taken from https://tikz.net/electric_field/ by Izaak Neutelings (July 2018) [^2]: Taken from https://tikz.net/electric_potential_plots/ by Izaak Neutelings (February 2020)