Editing Classical field theory (section)

== Non-relativistic field theories ==
Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was with [[Michael Faraday|Faraday's]] [[lines of force]] when describing the [[electric field]]. The [[gravitational field]] was then similarly described.

===Newtonian gravitation===
The first [[field theory (physics)|field theory]] of gravity was [[Newton's theory of gravitation]] in which the mutual interaction between two [[mass]]es obeys an [[inverse square law]]. This was very useful for predicting the motion of planets around the Sun.

Any massive body ''M'' has a [[gravitational field]] '''g''' which describes its influence on other massive bodies. The gravitational field of ''M'' at a point '''r''' in space is found by determining the force '''F''' that ''M'' exerts on a small [[test mass]] ''m'' located at '''r''', and then dividing by ''m'':<ref name="kleppner85">{{cite book|last1=Kleppner|first1=David|last2=Kolenkow|first2=Robert|title=An Introduction to Mechanics|page=85}}</ref>
<math display="block"> \mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m}.</math>
Stipulating that ''m'' is much smaller than ''M'' ensures that the presence of ''m'' has a negligible influence on the behavior of ''M''.

According to [[Newton's law of universal gravitation]], '''F'''('''r''') is given by<ref name="kleppner85" />
<math display="block">\mathbf{F}(\mathbf{r}) = -\frac{G M m}{r^2}\hat{\mathbf{r}},</math>
where <math>\hat{\mathbf{r}}</math> is a [[unit vector]] pointing along the line from ''M'' to ''m'', and ''G'' is Newton's [[gravitational constant]]. Therefore, the gravitational field of ''M'' is<ref name="kleppner85" />
<math display="block">\mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m} = -\frac{G M}{r^2}\hat{\mathbf{r}}.</math>

The experimental observation that inertial mass and gravitational mass are equal to unprecedented levels of accuracy leads to the identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the [[equivalence principle]], which leads to [[general relativity]].

For a discrete collection of masses, ''M<sub>i</sub>'', located at points, '''r'''<sub>''i''</sub>, the gravitational field at a point '''r''' due to the masses is
<math display="block">\mathbf{g}(\mathbf{r})=-G\sum_i \frac{M_i(\mathbf{r}-\mathbf{r_i})}{|\mathbf{r}-\mathbf{r}_i|^3} \,, </math>

If we have a continuous mass distribution ''ρ'' instead, the sum is replaced by an integral,
<math display="block">\mathbf{g}(\mathbf{r})=-G \iiint_V \frac{\rho(\mathbf{x})d^3\mathbf{x}(\mathbf{r}-\mathbf{x})}{|\mathbf{r}-\mathbf{x}|^3} \, , </math>

Note that the direction of the field points from the position '''r''' to the position of the masses '''r'''<sub>''i''</sub>; this is ensured by the minus sign. In a nutshell, this means all masses attract.

In the integral form [[Gauss's law for gravity]] is
<math display="block">\iint\mathbf{g}\cdot d \mathbf{S} = -4\pi G M</math>
while in differential form it is
<math display="block">\nabla \cdot\mathbf{g} = -4\pi G\rho_m </math>

Therefore, the gravitational field '''g''' can be written in terms of the [[gradient]] of a [[gravitational potential]] {{math|''φ''('''r''')}}:
<math display="block">\mathbf{g}(\mathbf{r}) = -\nabla \phi(\mathbf{r}).</math>
This is a consequence of the gravitational force '''F''' being [[conservative field|conservative]].

=== Electromagnetism ===
==== 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>

==== Magnetostatics ====
{{Main|Magnetostatics}}

A steady current ''I'' flowing along a path ''ℓ'' will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by ''I'' on a nearby charge ''q'' with velocity '''v''' is
<math display="block">\mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}),</math>
where '''B'''('''r''') is the [[magnetic field]], which is determined from ''I'' by the [[Biot–Savart law]]:
<math display="block">\mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\boldsymbol{\ell} \times d\hat{\mathbf{r}}}{r^2}.</math>

The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a [[magnetic vector potential|vector potential]], '''A'''('''r'''):
<math display="block"> \mathbf{B}(\mathbf{r}) = \nabla \times \mathbf{A}(\mathbf{r}) </math>

[[Gauss's law]] for magnetism in integral form is
<math display="block">\iint\mathbf{B}\cdot d\mathbf{S} = 0, </math>
while in differential form it is
<math display="block">\nabla \cdot\mathbf{B} = 0. </math>

The physical interpretation is that there are no [[magnetic monopole]]s.

==== Electrodynamics ====
{{Main|Electrodynamics}}

In general, in the presence of both a charge density ''ρ''('''r''', ''t'') and current density '''J'''('''r''', ''t''), there will be both an electric and a magnetic field, and both will vary in time. They are determined by [[Maxwell's equations]], a set of differential equations which directly relate '''E''' and '''B''' to the electric charge density (charge per unit volume) ''ρ'' and [[current density]] (electric current per unit area) '''J'''.<ref name="griffiths326">{{cite book |last=Griffiths |first=David |title=Introduction to Electrodynamics |edition=3rd |page=326 }}</ref>

Alternatively, one can describe the system in terms of its scalar and vector potentials ''V'' and '''A'''. A set of integral equations known as ''[[retarded potential]]s'' allow one to calculate ''V'' and '''A''' from ρ and '''J''',{{NoteTag|This is contingent on the correct choice of [[gauge fixing|gauge]]. ''φ'' and '''A''' are not uniquely determined by ''ρ'' and '''J'''; rather, they are only determined up to some scalar function ''f''('''r''', ''t'') known as the gauge. The retarded potential formalism requires one to choose the [[Lorenz gauge]].}} and from there the electric and magnetic fields are determined via the relations<ref name="wangsness469">{{cite book |last = Wangsness |first=Roald |title=Electromagnetic Fields |edition=2nd |page=469 }}</ref>
<math display="block"> \mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t}</math>
<math display="block"> \mathbf{B} = \nabla \times \mathbf{A}.</math>

=== Continuum mechanics ===
==== Fluid dynamics ====
{{Main|Fluid dynamics}}

Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation is a continuity equation, representing the conservation of mass
<math display="block">\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf u) = 0 </math>
and the [[Navier–Stokes equations]] represent the conservation of momentum in the fluid, found from Newton's laws applied to the fluid,
<math display="block">\frac {\partial}{\partial t} (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u + p \mathbf I) = \nabla \cdot \boldsymbol \tau + \rho \mathbf b </math>
if the density {{mvar|ρ}}, pressure {{mvar|p}}, [[deviatoric stress tensor]] {{mvar|'''τ'''}} of the fluid, as well as external body forces '''b''', are all given. The [[velocity field]] '''u''' is the vector field to solve for.

=== Other examples ===
In 1839, [[James MacCullagh]] presented field equations to describe [[reflection (physics)|reflection]] and [[refraction]] in "An essay toward a dynamical theory of crystalline reflection and refraction".<ref>[[James MacCullagh]] (1839) [https://archive.org/stream/collectedworks00maccuoft#page/144/mode/2up An essay toward a dynamical theory of crystalline reflection and refraction], ''Transactions, [[Royal Irish Academy]] 21''</ref>