Electrostatics next up previous contents
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Electrostatics

The governing equations of electrostatics are

$\displaystyle \boldsymbol{ E} = - \nabla V$ (102)

and

$\displaystyle \nabla \cdot \boldsymbol{ E} = 4 \pi \rho^e,$ (103)

where $ \boldsymbol{ E}$ is the electric field, $ V$ is the electric potential and $ \rho^e$ is the electric charge density. The electric field $ \boldsymbol{ E}$ is the force on a unit charge. For metals, it is linked to the current density $ \boldsymbol{ j}$ by the electric conductivity $ \sigma$ [5]:

$\displaystyle \boldsymbol{ j} = \sigma \boldsymbol{ E}.$ (104)

The resulting equation now reads

$\displaystyle \nabla \cdot (- \boldsymbol{ I} \cdot \nabla V) = 4 \pi \rho^e.$ (105)

Accordingly, by comparison with the heat equation, the correspondence in Table  (13) arises. Notice that the electrostatics equation is a steady state equation, and there is no equivalent to the heat capacity term.


Table 13: Correspondence between the heat equation and the equation for electrostatics.
heat electrostatics
T $ V$
$ \boldsymbol{ q}$ $ \boldsymbol{ E}$
$ q_n$ $ E_n = \frac{ j_n}{\sigma}$
$ \boldsymbol{\kappa}$ $ \boldsymbol{ I}$
$ \rho h$ $ 4 \pi \rho^e$
$ \rho c$ $ -$

An application of electrostatics is the potential drop technique for crack propagation measurements: a predefined current is sent through a specimen. Due to crack propagation the specimen section is reduced and its electric resistance increases. This leads to an increase of the electric potential across the specimen. A finite element calculation for the specimen can determine the relationship between the potential and the crack length. This calibration curve can be used to derive the actual crack length from potential measurements during the test.


next up previous contents
Next: Stationary groundwater flow Up: Types of analysis Previous: Irrotational incompressible inviscid flow   Contents
guido dhondt 2012-10-06