Editing Circulation (physics) (section)

== Uses ==

=== Kutta–Joukowski theorem in fluid dynamics ===
{{main|Kutta–Joukowski theorem}}

In fluid dynamics, the [[lift (force)|lift]] per unit span (L') acting on a body in a two-dimensional flow field is directly proportional to the circulation. Lift per unit span can be expressed as the product of the circulation Γ about the body, the fluid density <math>\rho</math>, and the speed of the body relative to the free-stream <math>v_{\infty}</math>:
<math display="block">L' = \rho v_{\infty} \Gamma</math>

This is known as the Kutta–Joukowski theorem.<ref name="K&S">{{cite book | author1=A.M. Kuethe | title=Foundations of Aerodynamics | author2=J.D. Schetzer | publisher=[[John Wiley & Sons]] | year=1959 | isbn=978-0-471-50952-3 | edition=2 | at=§4.11}}</ref>

This equation applies around airfoils, where the circulation is generated by ''airfoil action''; and around spinning objects experiencing the [[Magnus effect]] where the circulation is induced mechanically. In airfoil action, the magnitude of the circulation is determined by the [[Kutta condition]].<ref name="K&S" />

The circulation on every closed curve around the airfoil has the same value, and is related to the lift generated by each unit length of span. Provided the closed curve encloses the airfoil, the choice of curve is arbitrary.<ref name="JDA" />

Circulation is often used in [[computational fluid dynamics]] as an intermediate variable to calculate forces on an [[airfoil]] or other body.

=== Fundamental equations of electromagnetism ===
In electrodynamics, the [[Faraday's law of induction#Maxwell–Faraday equation|Maxwell-Faraday law of induction]] can be stated in two equivalent forms:<ref>{{Cite web | title=The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction | url=https://feynmanlectures.caltech.edu/II_17.html | access-date=2020-11-02 | website=feynmanlectures.caltech.edu}}</ref> that the curl of the electric field is equal to the negative rate of change of the magnetic field,
<math display="block">\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>

or that the circulation of the electric field around a loop is equal to the negative rate of change of the magnetic field flux through any surface spanned by the loop, by Stokes' theorem
<math display="block">\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = \iint_S \nabla\times\mathbf{E} \cdot \mathrm{d}\mathbf{S} =
 - \frac{\mathrm{d}}{\mathrm{d}t} \int_{S} \mathbf{B} \cdot \mathrm{d}\mathbf{S}.</math>

Circulation of a [[static magnetic field]] is, by [[Ampère's law]], proportional to the total current enclosed by the loop
<math display="block">\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S} = \mu_0 I_\text{enc}.</math>

For systems with electric fields that change over time, the law must be modified to include a term known as Maxwell's correction.