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[math]\mathbf{E}\left(\mathbf{r} \right )=\frac{1}{4\pi \epsilon_0}\int \frac{\mathbf{\hat{r'}}}{r' ^2}\rho \mathrm{d} \tau[/math]

[math]V\left(\mathbf{r} \right )=\frac{1}{4\pi \epsilon_0}\int \frac{\rho}{r'} \mathrm{d} \tau[/math]

[math]\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \nabla \times \mathbf{E} = 0[/math]

[math]\nabla ^2 V = -\frac{\rho}{\epsilon_0}[/math]

[math]V(\mathbf{r})=-\int_{\mathcal{O}}^{\mathbf{r}} \mathbf{E} \cdot \mathrm{d} \mathbf{l}[/math]

[math]\mathbf{E}=-\nabla V[/math]

General formulation

The equations in this section are given in SI units. Unlike the equations of mechanics (for example), Maxwell's equations are not unchanged in other unit systems. Though the general form remains the same, various definitions get changed and different constants appear at different places. Other than SI (used in engineering), the units commonly used are Gaussian units (based on the cgs system and considered to have some theoretical advantages over SI), Lorentz-Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics). See below for CGS-Gaussian units.

Two equivalent, general formulations of Maxwell's equations follow. The first separates bound charge and bound current (which arise in the context of dielectric and/or magnetized materials) from free charge and free current (the more conventional type of charge and current). This separation is useful for calculations involving dielectric or magnetized materials. The second formulation treats all charge equally, combining free and bound charge into total charge (and likewise with current). This is the more fundamental or microscopic point of view, and is particularly useful when no dielectric or magnetic material is present. More details, and a proof that these two formulations are mathematically equivalent, are given in section 4.

Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities. The definitions of terms used in the two tables of equations are given in another table immediately following.

Formulation in terms of free charge and current

Name	Differential form	Integral form
Gauss's law	[math]\nabla \cdot \mathbf{D} = \rho_f[/math]	[math]\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf D\;\cdot\mathrm{d}\mathbf A = Q_{f}(V)[/math]
Gauss's law for magnetism	[math]\nabla \cdot \mathbf{B} = 0[/math]	[math]\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A = 0[/math]
Maxwell–Faraday equation (Faraday's law of induction)	[math]\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}[/math]	[math]\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {\partial \Phi_{B,S}}{\partial t} [/math]
Ampère's circuital law (with Maxwell's correction)	[math]\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}[/math] [math][/math]	[math]\oint_{\partial S} \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{f,S} + \frac {\partial \Phi_{D,S}}{\partial t} [/math]

Formulation in terms of total charge and current

Name	Differential form	Integral form
Gauss's law	[math]\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}[/math]	[math]\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf E\;\cdot\mathrm{d}\mathbf A = \frac{Q(V)}{\varepsilon_0}[/math]
Gauss's law for magnetism	[math]\nabla \cdot \mathbf{B} = 0[/math]	[math]\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A = 0[/math]
Maxwell–Faraday equation (Faraday's law of induction)	[math]\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}[/math]	[math]\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {\partial \Phi_{B,S}}{\partial t} [/math]
Ampère's circuital law (with Maxwell's correction)	[math]\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ [/math]	[math]\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_S + \mu_0 \varepsilon_0 \frac {\partial \Phi_{E,S}}{\partial t} [/math]

The following table provides the meaning of each symbol and the SI unit of measure:

Definitions and units

Symbol	Meaning (first term is the most common)	SI Unit of Measure
[math]\mathbf{E} \ [/math]	electric field also called the electric field intensity	volt per meter or, equivalently, newton per coulomb
[math]\mathbf{B} \ [/math]	magnetic field also called the magnetic induction also called the magnetic field density also called the magnetic flux density	tesla, or equivalently, weber per square meter, volt-second per square meter
[math]\mathbf{D} \ [/math]	electric displacement field also called the electric induction also called the electric flux density	coulombs per square meter or equivalently, newton per volt-meter
[math]\mathbf{H} \ [/math]	magnetizing field also called auxiliary magnetic field also called magnetic field intensity also called magnetic field	ampere per meter
[math]\mathbf{\nabla \cdot}[/math]	the divergence operator	per meter (factor contributed by applying either operator)
[math]\mathbf{\nabla \times}[/math]	the curl operator
[math]\frac {\partial}{\partial t}[/math]	partial derivative with respect to time	per second (factor contributed by applying the operator)
[math]\mathrm{d}\mathbf{A}[/math]	differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S	square meters
[math] \mathrm{d} \mathbf{l} [/math]	differential vector element of path length tangential to the path/curve	meters
[math]\varepsilon_0 \ [/math]	permittivity of free space, also called the electric constant, a universal constant	farads per meter
[math]\mu_0 \ [/math]	permeability of free space, also called the magnetic constant, a universal constant	henries per meter, or newtons per ampere squared
[math]\ \rho_f \ [/math]	free charge density (not including bound charge)	coulombs per cubic meter
[math]\ \rho \ [/math]	total charge density (including both free and bound charge)	coulombs per cubic meter
[math]\mathbf{J}_f[/math]	free current density (not including bound current)	amperes per square meter
[math]\mathbf{J}[/math]	total current density (including both free and bound current)	amperes per square meter
[math]\,Q_f (V)[/math]	net free electric charge within the three-dimensional volume V (not including bound charge)	coulombs
[math]\,Q(V)[/math]	net electric charge within the three-dimensional volume V (including both free and bound charge)	coulombs
[math]\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l}[/math]	line integral of the electric field along the boundary ∂S of a surface S (∂S is always a closed curve).	joules per coulomb
[math]\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l}[/math]	line integral of the magnetic field over the closed boundary ∂S of the surface S	tesla-meters
[math]\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf E\;\cdot\mathrm{d}\mathbf A[/math]	the electric flux (surface integral of the electric field) through the (closed) surface [math]\partial V[/math] (the boundary of the volume V)	joule-meter per coulomb
[math]\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A[/math]	the magnetic flux (surface integral of the magnetic B-field) through the (closed) surface [math]\partial V[/math] (the boundary of the volume V)	tesla meters-squared or webers
[math]\iint_S \mathbf{B} \cdot \mathrm{d} \mathbf{A} = \Phi_{B,S}[/math]	magnetic flux through any surface S, not necessarily closed	webers or equivalently, volt-seconds
[math]\iint_S \mathbf{E} \cdot \mathrm{d} \mathbf{A} = \Phi_{E,S}[/math]	electric flux through any surface S, not necessarily closed	joule-meters per coulomb
[math]\iint_S \mathbf{D} \cdot \mathrm{d} \mathbf{A} = \Phi_{D,S}[/math]	flux of electric displacement field through any surface S, not necessarily closed	coulombs
[math]\iint_S \mathbf{J}_f \cdot \mathrm{d} \mathbf{A} = I_{f,s}[/math]	net free electrical current passing through the surface S (not including bound current)	amperes
[math]\iint_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} = I_{S}[/math]	net electrical current passing through the surface S (including both free and bound current)	amperes

Maxwell's equations are generally applied to macroscopic averages of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms (where they undergo quantum mechanical effects as well). It is only in this averaged sense that one can define quantities such as the permittivity and permeability of a material. At microscopic level, Maxwell's equations, ignoring quantum effects, describe fields, charges and currents in free space—but at this level of detail one must include all charges, even those at an atomic level, generally an intractable problem.