Curl of a Vector

In electromagnetism and fluid dynamics we often determine the curl of a vector quantity. This is a mathematical operation that follows set rules and is used to determine a field vector. What does curl mean physically?

In electromagnetism when rationalised SI units are used, the curl of the electric field vector (E) is equal to minus the partial time derivative of the magnetic induction vector (B) and the curl of the magnetic field vector (H) is equal to the sum of the current density (J) and the partial time derivative of the electric displacement vector (D). These relationships are known as the Faraday-Maxwell law and the Ampere-Maxwell law respectively.  In fluid mechanics the curl of the fluid velocity vector (u) is equal to the vorticity (W).

The curl operation is a mathematical way of determining whether the work done (also called the circulation) in moving around a small circle centred at the field point is zero. If the work done is zero the field is said to be conservative or irrotational, meaning that a small paddle wheel placed at that point in the field would not spin if the field was the velocity field of a fluid, hence the name of the operation curl. Some textbooks call the curl operation rot, the significance coming from the rotation of the test paddle wheel. The mathematical rule connecting curl with work done is known as Stokes' theorem, one of the very important theorems of vector analysis along with those of Gauss and Green.

The curl operation is a very important part of the language of electromagnetism. The curl operator is part of the mathematical language that we use to show that time varying electric and magnetic fields propagate through space at the speed of light.

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