9/16/2005 The Curl of a Vector Field.doc 1/7
The Curl of a Vector Field
Say ∇=xrAB() () r. The mathematical definition of Curl is given as:
v∫ A(r ) ⋅d A Ci Bi ()rlim= ∆→s 0 ∆si
This rather complex equation requires some explanation !
* Bi ()r is the scalar component of vector B (r ) in the direction ˆ ˆˆˆ defined by unit vector ai (e.g., aaax ,,ρ θ ).
* The small surface ∆si is centered at point r , and oriented ˆ such that it is normal to unit vector ai .
* The contour Ci is the closed contour that surrounds surface ∆s . i C i
ˆai
r ∆si
Jim Stiles The Univ. of Kansas Dept. of EECS 9/16/2005 The Curl of a Vector Field.doc 2/7
Note that this derivation must be completed for each of the three orthonormal base vectors in order to completely define BA()rxr=∇ ().
Q: What does curl tell us ?
A: Curl is a measurement of the circulation of vector field A ()r around point r .
If a component of vector field A (r )is pointing in the direction
d A at every point on contour Ci (i.e., tangential to the contour). Then the line integral, and thus the curl, will be positive.
If, however, a component of vector field A (r ) points in the opposite direction (-d A) at every point on the contour, the curl at point r will be negative.
< Bi 0 Bi > 0
Likewise, these vector fields will result in a curl with zero value at point r :
Jim Stiles The Univ. of Kansas Dept. of EECS 9/16/2005 The Curl of a Vector Field.doc 3/7
B = 0 Bi = 0 i * Generally, the curl of a vector field result is in another vector field whose magnitude is positive in some regions of space, negative in other regions, and zero elsewhere.
* For most physical problems, the curl of a vector field provides another vector field that indicates rotational sources (i.e., “paddle wheels” ) of the original vector field.
For example, consider this vector field A()r :
4
2
y 0 x
-2
-4
- 4 - 2 0 2 4
Jim Stiles The Univ. of Kansas Dept. of EECS 9/16/2005 The Curl of a Vector Field.doc 4/7
If we take the curl of A()r , we get a vector field which points in the direction aˆz at all points (x,y). The scalar component of this resulting vector field (i.e., Bz(r )) is:
4
2
y 0 0.75 4 -2 0.5 0.25 Ax x,y 2 0 -4 H L - 4 0 y - 2 -4 -2 0 2 4 x - 2 0
x 2 - 4 4
The relationship between the original vector field A()r and its resulting curl perhaps is best shown when plotting both together:
4
2
y 0
- 2
- 4
- 4 - 2 0 2 4 x
Jim Stiles The Univ. of Kansas Dept. of EECS 9/16/2005 The Curl of a Vector Field.doc 5/7
Note this scalar component is largest in the region near point x=-1, y=1, indicating a “rotational source” in this region. This is likewise apparent from the original plot of vector field A()r .
Consider now another vector field:
4
2
0 y x
-2
-4
-4 - 2 0 2 4
Although at first this vector field appears to exhibit no rotation, it in fact has a non-zero curl at every point
(B()r = 4.0aˆz ) ! Again, the direction of the resulting field is in the direction aˆz . We plot therefore the scalar component in this direction (i.e., Brz ()): y 4 2 0 2 4 - -4
2 8
6
y 0 4 Ax x,y 2 -2 0 H L -4 -4 -2 0
-4 - 2 0 2 4 2 x x 4
Jim Stiles The Univ. of Kansas Dept. of EECS 9/16/2005 The Curl of a Vector Field.doc 6/7
We might encounter a more complex vector field, such as:
4
2
0
-2
-4
-4 - 2 0 2 4
If we take the curl of this vector field, the resulting vector field will again point in the direction aˆz at every point (i.e.,
Brxy()== Br () 0). Plotting therefore the scalar component of the resulting vector field (i.e., Brz ()), we get:
4
2
y 0 1 4 0 Ax x,y 2 - 1 -2 H L - 4 0 y - 2 - 2 0 -4 x 2 - 4 4 -4 -2 0 2 4 x
Note these plots indicate that there are two regions of large counter clockwise rotation in the original vector field, and one region of large clockwise rotation.
Jim Stiles The Univ. of Kansas Dept. of EECS 9/16/2005 The Curl of a Vector Field.doc 7/7
4
2
y 0
-2
-4
-4 -2 0 2 4 x
Finally, consider these vector fields:
4 4
2 2
0 0
- 2 - 2
- 4 - 4
- 4 - 2 0 2 4 - 4 - 2 0 2 4
The curl of these vector fields is zero at all points. It is apparent that there is no rotation in either of these vector fields!
Jim Stiles The Univ. of Kansas Dept. of EECS