> [!summary] Capacitance is the measure of charge per volt in a capacitor > **Key equations:** > Capacitance: $C = \frac{Q}{V}$ > Capacitance for parallel plates: $C=\frac{\epsilon_0A}{d}$ >[!info]+ Read Time **⏱ 2 mins** # Definition Capacitance is a measure of how much charge a [[Capacitors|capacitor]] can hold per [[Volt|volt]]. Volts are the [[Potential Difference|potential difference]] between the plates of a [[Capacitors|capacitor]]. $ C = \frac{Q}{V} $ # Deriving Capacitance for Parallel Plates >[!warning] Assumptions To find the capacitance, assume that the plates are even and symmetric. Allowing arguments from [[Gauss's Law|Gauss's law]]. $\begin{array}{c} \text{Use Gauss's Law arguments for two plates} \\ \oint \vec{E}\cdot \vec{dA} = \frac{Q_{enc}}{\epsilon_0} \\ \text{Well assume our Gaussian surface as some closed square with area A. } \\ \text{Well use this same argument for finding the Q} \\ \\ Q_{enc} = \lambda A \\\\ \text{Where our charge density $\lambda$ is defined through the area. Solving for the electric field} \\\\ \oint \vec{E}\cdot \vec{dA} = \frac{Q_{enc}}{\epsilon_0} \\ \vec{E} \cdot A= \frac{\lambda A}{\epsilon_0} \\ \vec{E} = \frac{\lambda}{\epsilon_0} \end{array}$ >[!bug] Note > Recall the equation from [[Electric Potential]] ($\Delta V=\frac{\Delta U}{q_0}=-\int_a^b E\cdot ds$) > We will want our $Q$ in terms of the electric field (for the later part of this derivation)so we will solve for $Q_{enc}$ in terms of the electric field. $\begin{array}{c} EA = \frac{Q_{enc}}{\epsilon_0} \\ Q_{enc} = EA\epsilon_0 \\ \\ \text{So then our electric potential is} \\\\ \Delta V = -\int_{a}^b \vec{E}\cdot\vec{ds} \\ \end{array}$ >[!bug] Note > Assume that the [[Electric Fields|electric field]] is constant through a to b and total distance is $d$ We're going to assume our starting point is from the positive plate to negative plate, but our displacement starts from the negative plate to positive. $ \begin{array}{c} \\ \text{Now our electric potential is:} \\ V = \vec{E} d \\ \\ \text{Now using our defintion of capacitance:}\\ C=\frac{Q}{V}=\frac{EA\epsilon_0}{Ed}=\frac{\epsilon_0A}{d} \end{array} $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/f_MZNsEqyQw?si=wlkZmT4pyZPcSny0" 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> --- > 📚 Like this note? [Star the GitHub repo](https://github.com/rajeevphysics/Obsidian-MathMatter) to support the project and help others discover it! ---