# Pre-Quantum Electrodynamics
Author: Jean-Sébastien Caux

##### Capacitors
ems.es.c.cap

Two conductors, one at charge \(Q\) and the other at charge \(-Q\). Potential difference is unambiguous:

\[ V = V_+ - V_- = -\int_{(-)}^{(+)} {\bf E} \cdot d{\bf l} \]

Capacitance

\[ C \equiv \frac{Q}{V} \label{Gr(2.53)} \]

Example 2.11: find the capacitance of two concentric spherical metallic shells with radii \(a\) and \(b\).

\[ V = -\int_b^a {\bf E} \cdot d{\bf l} = \frac{Q}{4\pi \varepsilon_0} (\frac{1}{a} - \frac{1}{b}), \rightarrow C = \frac{Q}{V} = 4\pi \varepsilon_0 \frac{ab}{b - a}. \]

Work to charge up capacitor: \(dW = \frac{q}{C} dq\). Integrating,

\[ W = \frac{1}{2} C V^2 \label{Gr(2.55)} \]

Author: Jean-Sébastien Caux

Created: 2022-02-08 Tue 06:55

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