derivation of Coulomb’s Law from Gauss’ Law

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. We can obtain an expression for the electric field surrounding the charge. We surround the charge with a ”virtual” sphere of radius $R$, then use Gauss’ law in integral form:

$${\oint }_S𝐄\cdot d𝐀=\frac{q}{{ϵ}_0}$$

We rewrite this as a volume integral in spherical polar coordinates over the ”virtual” sphere mentioned above, which has the point charge at its centre. Since the electric field is spherically symmetric (by assumption) the electric field is constant over this volume.

$${\oint }_S𝐄\cdot d𝐀={\int }_0^R{\int }_0^{2\pi }{\int }_0^{\pi }Er\sin \theta drd\theta d\varphi$$

Hence

$$4\pi R^{2}E=\frac{q}{{ϵ}_0}$$

Or

$$E=\frac{q}{4\pi {ϵ}_0{R}^2}$$

The usual form can then be recovered from the Lorentz force law, $𝐅=𝐄q+𝐯\times 𝐁$ noting the absence of magnetic field.

Title	derivation of Coulomb’s Law from Gauss’ Law
Canonical name	DerivationOfCoulombsLawFromGaussLaw
Date of creation	2013-03-22 17:53:23
Last modified on	2013-03-22 17:53:23
Owner	invisiblerhino (19637)
Last modified by	invisiblerhino (19637)
Numerical id	8
Author	invisiblerhino (19637)
Entry type	Derivation
Classification	msc 35Q60
Classification	msc 78A25
Defines	Coulomb’s Law