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[parent] derivation of Coulomb's Law from Gauss' Law (Derivation)

As an example of the statement that Maxwell's equations completely define electromagnetic phenomena, it will be shown that Coulomb's law may be derived from Gauss' Law for electrostatics. Consider a point charge of charge $q$; we can obtain an expression for the Electric Field at a point in space due to this charge by surrounding it with a "virtual" sphere of radius $R$, and then using the Gauss' law in integral form:

$\displaystyle \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac {q}{\epsilon_0}$ .
The surface integral on the the right-hand-size of the equation can be written in spherical polar coordinates over the "virtual" sphere, considering the point charge at its centre. Under the assumption that the electric field is spherically symmetric, its value over the sphere surface is constant. Then, we can write

$\displaystyle \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \int^{2\pi}_0 \int^\pi_0 R^2 E \sin \theta \,d\theta \,d\phi; $
hence,

$\displaystyle 4\pi R^2 E = \frac{q}{\epsilon_0}, $
or

$\displaystyle E = \frac{q}{4\pi\epsilon_0 R^2}. $
The usual form can then be recovered from the Lorentz force law $\mathbf{F} = \mathbf{E}q + \mathbf{v} \times \mathbf{B}$, noting the absence of magnetic field.



"derivation of Coulomb's Law from Gauss' Law" is owned by victor_phb. [ full author list (2) | owner history (1) ]

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See Also: Maxwell's equations


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Cross-references: magnetic field, Lorentz force law, Electric Field, charge, Gauss Law, Coulomb's law, Maxwell's equations

This is version 2 of derivation of Coulomb's Law from Gauss' Law, born on 2008-03-20, modified 2011-02-03.
Object id is 272, canonical name is DerivationOfCoulombsLawFromGaussLaw.
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Physics Classification: 40. (ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS)
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