Pre-Quantum Electrodynamics
Capacitorsems.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,
\[ W = \frac{1}{2} C V^2 \label{Gr(2.55)} \]
Created: 2022-02-08 Tue 06:55