Pre-Quantum Electrodynamics

Capacitors ems.ca.c.cap
  • PM 3.5, 3.6, 3.7
  • Gr 2.5.4

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

  • PM (3.7)
  • Gr (2.53)

\[ C \equiv \frac{Q}{V} \]

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,

  • PM (3.29)
  • Gr (2.55)

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



Creative Commons License Except where otherwise noted, all content is licensed under a Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

Created: 2024-02-27 Tue 10:31