Pre-Quantum Electrodynamics
Surface Charge and the Force on a Conductorems.ca.c.sc
- PM 3.4
- Gr 2.5.3
For a conductor, we can exploit the fact that electrical fields vanish on the inside to get a proper boundary condition for the potential. Namely, here, boundary condition Edisc yields
\[ {\bf E} = \frac{\sigma}{\varepsilon_0} \hat{\bf n} \tag{E_sur_cond}\label{E_sur_cond} \]
which in terms of potential reads
\[ \sigma = -\varepsilon_0 \frac{\partial \phi}{\partial n}. \tag{scd_cond}\label{scd_cond} \]
The force per unit area on the surface of an object is
- Gr (2.50)
\[ {\bf f} = \sigma {\bf E}_{\mbox{average}} = \frac{\sigma}{2} ({\bf E}_{\mbox{above}} + {\bf E}_{\mbox{below}}) \]
For a conductor, \({\bf E}_{\mbox{below}} = 0\), \({\bf E}_{\mbox{above}} = \frac{\sigma}{\varepsilon_0} \hat{\bf n}\), so
- Gr (2.51)
\[ {\bf f} = \frac{\sigma^2}{2\varepsilon_0} \hat{\bf n} \] amounting to an outward electrostatic pressure. In terms of the field,
- Gr (2.52)
\[ P = \frac{\varepsilon_0}{2} E^2 \] which can also be obtained from the principle of virtual work (considering the change rate of energy with respect to volume).

Created: 2024-02-27 Tue 10:31