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Curl of a Vector

Dr. Rajib Kumar Panigrahi

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Curl

The curl of A is an axial (or rotational) vector whose

magnitude is the maximum circulation of Aper unit area asthe area tends to zero and whose direction is the normal

direction of the area when the area is oriented so as to make

the circulation maximum.

where the area S is bounded by the curve Land anis the unit vector normal to the surfaceS.

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Meaning of Curl

The curlof the wind vectorfield F, curl F, measures itsspinning effect.

It is a vector field thatlines up with the axis alongwhich the wind is trying totwirl you, and whosemagnitude indicates the

strength of the twirlingeffect, in the counter-clockwise direction. Fig.: The paddel wheel interpretation

of curl F

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Note this scalar component is largest inthe region near point x=-1, y=1, indicating

a rotational source in this region.

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Curl

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AMPERE'S CIRCUIT LAW

Ampere's circuit law states that the line integralof the tangential component of H around a

closed path is the same as the net current Ienc

enclosed by the path. In other words, the

circulation of HequalsIenc; that is,

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L

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The vector field F= -y ax+x ay

( ) ( ) 0y xx y

F Curl F= 2az

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Vector Field F = y axx ay

Curl F= -2az

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The Vector Field v(x,y) = (y axx ay) / (x2+y2)

Curl v= 0 !!

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Stokes's theoremStokes's theorem states that the circulation of a vector field A

around a (closed) pathLis equal to the surface integral of the

curl of Aover the open surface S bounded by Lprovided that

Aand Aare continuous on S.

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Properties of Curl

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If A= r cosar+ sin a, evaluate

around the path as shown in the Figure. Confirm

this using Stokes's theorem.

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CLASSIFICATION OF VECTOR FIELDS

A vector field is uniquely characterized by its

divergence and curl.

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A vector field Ais said to be solenoidal (or divergenceless)

if A= 0. Such a field has neither source nor sink of flux. From the

divergence theorem,

Examples of solenoidal fields are incompressible fluids,

magnetic fields, and conduction current density under

steady state conditions.

In general, the field of curl F(for any F) is purely

solenoidal because (

F) = 0.

Solenoidal (or divergenceless) field

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Irrotational field A vector field Ais said to be irrotational (or potential) if

A= 0.

From Stokes's theorem,

Examples of irrotational fields include the electrostatic field

and the gravitational field.

In general, the field of gradient V (for any scalar V) ispurely irrotational since (

V) = 0

Thus, an irrotational field Acan always be expressed in

terms of a scalar field V; that is A = -

V.