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

Let us consider two conductors, one having charge \(Q\) and the other charge \(-Q\). The potential difference is unambiguously given by

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

Capacitance

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

Concentric shells

Consider two concentric spherical metallic shells with radii \(a\) and \(b\). What is the capacitance of this system?

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

Work to charge up capacitor: start from the differential work to add charge \(dq\) to the already-present charge \(q\): \(dW = \frac{q}{C} dq\). Integrating,

\[ W = \frac{1}{2} C V^2 \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42