Editing Classical field theory (section)

==== Electrostatics ====
{{Main|Electrostatics}}

A [[test charge|charged test particle]] with charge ''q'' experiences a force '''F''' based solely on its charge. We can similarly describe the [[electric field]] '''E''' generated by the source charge ''Q'' so that {{math|1='''F''' = ''q'''''E'''}}:
<math display="block"> \mathbf{E}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{q}.</math>

Using this and [[Coulomb's law]] the electric field due to a single charged particle is
<math display="block">\mathbf{E} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2} \hat{\mathbf{r}} \,. </math>

The electric field is [[conservative field|conservative]], and hence is given by the gradient of a scalar potential, {{math|''V''('''r''')}}
<math display="block"> \mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r}) \, . </math>

[[Gauss's law]] for electricity is in integral form
<math display="block">\iint\mathbf{E}\cdot d\mathbf{S} = \frac{Q}{\varepsilon_0}</math>
while in differential form
<math display="block">\nabla \cdot\mathbf{E} = \frac{\rho_e}{\varepsilon_0} \,. </math>