# Flux
**Flux** is a measure of how much of a vector field passes through a surface, defined as the [surface integral](https://www.edgepedia.org/surface-integral) of the component of the field perpendicular to the surface. In electromagnetism the two central cases are electric flux, the surface integral of the electric field \( \overrightarrow{E} \), and magnetic flux, the surface integral of the magnetic flux density \( \overrightarrow{B} \). The word generalizes the everyday idea of flow: for a velocity field describing a liquid, the flux through a closed surface is the net amount of fluid leaving per unit time, the average outward normal component of the velocity multiplied by the surface area.[1](https://www.feynmanlectures.caltech.edu/II_01.html)
| Key fact | Detail |
|---|---|
| Definition | Flux of a vector field through a surface is the surface integral of the normal component of the vector; for a uniform field and flat area, \( \Phi = \overrightarrow{E} \cdot \overrightarrow{A} = EA\cos\theta \)[2](https://openstax.org/books/university-physics-volume-2/pages/6-key-equations) • [1](https://www.feynmanlectures.caltech.edu/II_01.html) |
| Electric flux, closed surface | \( \Phi = \oint_S \overrightarrow{E} \cdot \widehat{n}\,dA = q_{\text{enc}}/\epsilon_0 \) (Gauss's law)[2](https://openstax.org/books/university-physics-volume-2/pages/6-key-equations) |
| Magnetic flux | \( \Phi_m = \int_S \overrightarrow{B} \cdot \widehat{n}\,dA \); SI unit the weber (Wb), with 1 Wb = 1 T·m² = 1 V·s[3](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) |
| Faraday's law | Induced emf \( \epsilon = -d\Phi_m/dt \); for a coil of N tightly wound turns, \( \epsilon = -N\,d\Phi_m/dt \)[3](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) |
| Magnetic flux, closed surface | The total magnetic flux through any closed surface is zero[1](https://www.feynmanlectures.caltech.edu/II_01.html) |
| Mathematical tool | The divergence theorem converts a flux integral over a closed surface into a volume integral of the divergence[4](https://openstax.org/books/calculus-volume-3/pages/6-8-the-divergence-theorem) |
## Definition
The flux of a vector field through a surface is the surface integral of the normal component of the vector. Feynman's Lectures on Physics, the Caltech course by [Richard P. Feynman](https://www.edgepedia.org/richard-p-feynman), introduce the idea with heat flow, where the vector \( \overrightarrow{h} \) represents heat flowing through a unit area per unit time, and then generalize: even when the vector does not represent the flow of anything, as with the electric field, the surface integral of its normal component is still called the flux.[5](https://www.feynmanlectures.caltech.edu/II_03.html) For a uniform field and a flat surface of area A, the integral reduces to a dot product, \( \Phi = EA\cos\theta \), where \( \theta \) is the angle between the field and the surface normal.[2](https://openstax.org/books/university-physics-volume-2/pages/6-key-equations) The dot product means that only the component of the field perpendicular to the surface contributes; a field parallel to the surface gives zero flux, and a field perpendicular to it gives the maximum, \( EA \).[6](https://courses.physics.ucsd.edu/2012/Spring/physics1b/Electricity_1B.pdf)
The surface over which the integral is taken may be open or closed. For a closed surface the integral sign carries a circle, \( \oint \), and the normal \( \widehat{n} \) is conventionally taken to point outward.[2](https://openstax.org/books/university-physics-volume-2/pages/6-key-equations) For an open surface bounded by a curve, the direction of the normal is fixed by a right-hand rule relating it to the orientation of the boundary curve, the same convention used in [Stokes' theorem](https://www.edgepedia.org/stokes-theorem).[7](https://phys.libretexts.org/Courses/Berea_College/Electromagnetics_I_%28Messina%29/08%3A_Time-Varying_Fields/8.03%3A_Faradays_Law)
## Electric flux and Gauss's law
[Electric flux](https://www.edgepedia.org/electric-flux) is the surface integral of the electric field. Its physical content comes from [Gauss's law](https://www.edgepedia.org/gausss-law), one of the two laws of electrostatics: the flux of \( \overrightarrow{E} \) through any closed surface equals the net charge inside that surface divided by \( \epsilon_0 \), the permittivity of free space.[8](https://www.feynmanlectures.caltech.edu/II_04.html) In symbols,
\[ \oint_S \overrightarrow{E} \cdot \widehat{n}\,dA = \frac{q_{\text{enc}}}{\epsilon_0}. \]
The field \( \overrightarrow{E} \) in this integral is the total field at each point of the surface, including contributions from charges outside the surface, but only the enclosed charge appears on the right-hand side.[9](https://openstax.org/books/fizyka-dla-szk%C3%B3%C5%82-wy%C5%BCszych-tom-2/pages/6-2-wyjasnienie-prawa-gaussa) Charges outside a closed surface contribute zero net flux, because field lines that enter the surface also leave it.[1](https://www.feynmanlectures.caltech.edu/II_01.html) For a continuous charge distribution, the enclosed charge is found by integrating the charge density over the volume inside the surface.[8](https://www.feynmanlectures.caltech.edu/II_04.html)
**Radius independence.** For a point charge surrounded by a spherical [Gaussian surface](https://www.edgepedia.org/gaussian-surface), the flux works out to \( q/\epsilon_0 \) regardless of the sphere's radius: the field magnitude falls as \( 1/r^2 \) while the sphere's area grows as \( r^2 \), so their product is constant.[9](https://openstax.org/books/fizyka-dla-szk%C3%B3%C5%82-wy%C5%BCszych-tom-2/pages/6-2-wyjasnienie-prawa-gaussa) This is the same calculation that recovers Coulomb's inverse-square law from Gauss's law plus spherical symmetry.[1](https://www.feynmanlectures.caltech.edu/II_01.html)
**Using symmetry.** Gauss's law is most useful for calculating field magnitudes when the charge distribution has high symmetry. The method requires choosing a Gaussian surface on which the field is parallel to the area vector and uniform in magnitude, so the field can be pulled out of the integral, leaving \( EA = q_{\text{enc}}/\epsilon_0 \).[10](https://lipa.physics.oregonstate.edu/sec_gauss.html) The flux itself does not depend on the shape of the surface, but a well-chosen shape simplifies the calculation.[10](https://lipa.physics.oregonstate.edu/sec_gauss.html) One standard result obtained this way is the magnitude of the electric field just outside the surface of a conductor, \( E = \sigma/\epsilon_0 \), where \( \sigma \) is the local surface charge density.[2](https://openstax.org/books/university-physics-volume-2/pages/6-key-equations)
## Magnetic flux and induction
[Magnetic flux](https://www.edgepedia.org/magnetic-flux) is the surface integral of the magnetic flux density \( \overrightarrow{B} \),
\[ \Phi_m = \int_S \overrightarrow{B} \cdot \widehat{n}\,dA. \]
It measures the amount of magnetic field lines passing through a given surface area, a definition directly analogous to electric flux.[3](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) The SI unit is the weber (Wb), with 1 Wb = 1 T·m²; the weber is also a volt-second, and magnetic field strength is occasionally expressed in webers per square meter instead of teslas.[3](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) For a uniform field, \( \Phi = BA\cos\theta \), where \( B\cos\theta \) is the field component perpendicular to the area.[11](https://openstax.org/books/physics/pages/20-3-electromagnetic-induction)
Unlike electric flux, magnetic flux through a closed surface carries no source term: the total magnetic flux through any closed surface is zero, because no magnetic charges have been found.[1](https://www.feynmanlectures.caltech.edu/II_01.html)
**Faraday's law.** The practical importance of magnetic flux comes from electromagnetic induction. [Michael Faraday](https://www.edgepedia.org/michael-faraday)'s experiments, carried out mostly in 1831, showed that a current appears in a circuit only when the magnetic field it experiences is changing; the effect was discovered independently by [Joseph Henry](https://www.edgepedia.org/joseph-henry).[3](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) • [12](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/06%3A_Electromagnetism/6.01%3A_Electromagnetic_Induction) Faraday's law states that the induced emf is the negative rate of change of the magnetic flux,
\[ \epsilon = -\frac{d\Phi_m}{dt}. \]
The flux can change because the field changes, because the circuit moves or deforms, or because its orientation changes; any of these induces a voltage.[12](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/06%3A_Electromagnetism/6.01%3A_Electromagnetic_Induction) The negative sign gives the direction of the induced current, described by [Lenz's law](https://www.edgepedia.org/lenzs-law): the magnetic field of the induced current partially compensates the change in the original flux.[3](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) • [12](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/06%3A_Electromagnetism/6.01%3A_Electromagnetic_Induction) In a coil of N tightly wound turns, each turn encloses the same flux, so the induced emf is N times that of a single turn, \( \epsilon = -N\,d\Phi_m/dt \).[3](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law)
The surface used to evaluate the flux is arbitrary: any open surface bounded by the circuit gives the same value of \( \Phi_m \), whether planar or extending above and below the plane of the loop.[3](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) • [7](https://phys.libretexts.org/Courses/Berea_College/Electromagnetics_I_%28Messina%29/08%3A_Time-Varying_Fields/8.03%3A_Faradays_Law)
## Flux in the structure of electromagnetism
Flux and its companion idea, circulation, the line integral of the tangential component around a closed curve, are enough to state all the laws of electricity and magnetism. The flux of \( \overrightarrow{E} \) through any closed surface gives Gauss's law; the circulation of \( \overrightarrow{E} \) around a closed curve equals the negative rate of change of the magnetic flux through any surface bounded by that curve, which is Faraday's law in integral form.[1](https://www.feynmanlectures.caltech.edu/II_01.html) Applying Stokes' theorem to the differential form of Faraday's law, \( \nabla \times \overrightarrow{E} = -\partial\overrightarrow{B}/\partial t \), yields the same integral statement.[13](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_%28Staelin%29/02%3A_Introduction_to_Electrodynamics/2.04%3A_Relation_between_integral_and_differential_forms_of_Maxwell%E2%80%99s_equations)
The [divergence theorem](https://www.edgepedia.org/divergence-theorem) underlies the flux form of the laws: it states that the surface integral of the normal component of an arbitrary vector over a closed surface equals the volume integral of the vector's divergence over the enclosed volume.[5](https://www.feynmanlectures.caltech.edu/II_03.html) Applied to the electric field, it converts Gauss's law between its integral and differential forms, and it explains why the flux through a closed surface is \( q/\epsilon_0 \) when the charge is inside and zero when it is outside.[4](https://openstax.org/books/calculus-volume-3/pages/6-8-the-divergence-theorem)
## References
1. [The Feynman Lectures on Physics Vol. II Ch. 1: Electromagnetism, R. P. Feynman, Caltech](https://www.feynmanlectures.caltech.edu/II_01.html)
2. [Ch. 6 Key Equations, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/6-key-equations)
3. [13.1 Faraday's Law, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law)
4. [6.8 The Divergence Theorem, Calculus Volume 3, OpenStax](https://openstax.org/books/calculus-volume-3/pages/6-8-the-divergence-theorem)
5. [The Feynman Lectures on Physics Vol. II Ch. 3: Vector Integral Calculus, R. P. Feynman, Caltech](https://www.feynmanlectures.caltech.edu/II_03.html)
6. [Physics 1B: Electricity and Magnetism, UC San Diego](https://courses.physics.ucsd.edu/2012/Spring/physics1b/Electricity_1B.pdf)
7. [8.3: Faraday's Law, Physics LibreTexts](https://phys.libretexts.org/Courses/Berea_College/Electromagnetics_I_%28Messina%29/08%3A_Time-Varying_Fields/8.03%3A_Faradays_Law)
8. [The Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics, R. P. Feynman, Caltech](https://www.feynmanlectures.caltech.edu/II_04.html)
9. [6.2 Wyjaśnienie prawa Gaussa, Fizyka dla szkół wyższych. Tom 2, OpenStax](https://openstax.org/books/fizyka-dla-szk%C3%B3%C5%82-wy%C5%BCszych-tom-2/pages/6-2-wyjasnienie-prawa-gaussa)
10. [Gauss's Law, Oregon State University Physics](https://lipa.physics.oregonstate.edu/sec_gauss.html)
11. [20.3 Electromagnetic Induction, Physics, OpenStax](https://openstax.org/books/physics/pages/20-3-electromagnetic-induction)
12. [6.1: Electromagnetic Induction, Essential Graduate Physics, Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/06%3A_Electromagnetism/6.01%3A_Electromagnetic_Induction)
13. [2.4: Relation between integral and differential forms of Maxwell's equations, Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_%28Staelin%29/02%3A_Introduction_to_Electrodynamics/2.04%3A_Relation_between_integral_and_differential_forms_of_Maxwell%E2%80%99s_equations)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Maxwell's equations and field formulations › Maxwell's equations and potentials*
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