Revision as of 20:18, 6 December 2020 by Denneyl (Talk | contribs)

The Laplace operator has many applications in the physical sciences, one of which being in electric potentials. An electric field, $ E $, is defined as a vector field that describes the force of electricity per unit charge on any charge in the field. Take, for example, an electric field created by a point charge at $ (0,0) $. By Coulomb's law:

$ E = \frac{F}{q} \\ F = \large\frac{Qq}{4\pi\epsilon_{0}r^{2}} \\ E = \large\frac{Qq}{4\pi\epsilon_{0}r^{2}} \cdot \frac{1}{q} \\ E = \large\frac{Q}{4\pi\epsilon_{0}r^{2}} $

where $ Q $ is the charge of the point charge, $ q $ is the charge of a charge in the field, $ r $ is the distance of the charge from the point charge, and $ \epsilon_0 $ is vacuum permittivity, a physical constant approximately equal to $ 8.8 $ x $ 10^{-12} $ Farads per meter. If the point charge is 1 Coulomb, a 1 Coulomb charge 1 meter away from the point charge would have an electric force of $ \frac{1}{4\pi\epsilon_0} $ Newtons.

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