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[[Category:MA271Fall2020Walther]] | [[Category:MA271Fall2020Walther]] | ||
| + | The Laplace operator has many applications in the physical sciences, one of which being in electric potentials. An electric field, <math>E</math>, 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 <math>(0,0)</math>. By Coulumb's law: | ||
| + | |||
| + | <math> | ||
| + | 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}} | ||
| + | </math> | ||
| + | |||
| + | where <math>Q</math> is the charge of the point charge, <math>q</math> is the charge of a charge in the field, <math>r</math> is the distance of the charge from the point charge, and <math>\epsilon_0</math> is vacuum permittivity, a physical constant approximately equal to <math>8.8</math> x <math>10^{-12}</math> Farads per meter. | ||
<math></math> | <math></math> | ||
[[Walther_MA271_Fall2020_topic9|Back to main page]] | [[Walther_MA271_Fall2020_topic9|Back to main page]] | ||
Revision as of 19:09, 6 December 2020
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 Coulumb'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.
