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Circulation & Curl

Dr Aiden Price

EGB241

Week 10

1/14

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

2/14

Introduction

• So far, we have described the gradient of a scalar and the

divergence of a vector.

• This lecture will introduce the curl of a vector field.

• First, begin by revisiting theory we will now describe as

the circulation of a vector field.

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

3/14

Circulation of a Vector Field

• The curl of a vector field, B, describes the rotational

property, or the circulation, of B.

• For a closed contour, C, the circulation of B is defined as

the line integral of B around C. That is,

Circulation = IC B · dl.

• Two examples can be used to gain some physical

understanding. The second will be quite familiar.

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

4/14

Circulation Cases I & II

Case I: Case II:

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

5/14

Circulation – Case I

• The image (a) depicts uniform field, B = B0xˆ.

• For the rectangular contour, abcd, in image (a), the

circulation is calculated below.

Circulation = Zab B0xˆ · xˆ dx + Zbc B0xˆ · yˆ dy

+ Zcd B0xˆ · xˆ dx + Zda B0xˆ · yˆ dy

= B0∆x + 0 – B0∆x + 0

= 0.

• Thus, the circulation of a uniform field is zero.

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

6/14

Circulation – Case II

• The image (b) depicts a magnetic field, B, induced by an

infinite wire carrying a d-c current, I.

• If the current is in free space and it is oriented along the

z-direction, then

B = µ0I

2πr

ˆ φ.

• Where µ0 is the permeability of free space and r is the

radial distance from the current in the x – y plane. The

circulation is calculated below.

Circulation = Z02π 2µπr 0I φˆ · φr dφ ˆ

= µ0I

• Remember: Circulation depends on choice of contour.

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

7/14

Curl I

• The curl operator acts as a form of differentiation for

vector fields.

• The curl of B the circulation of B per unit area.

∇ × B = Curl B = lim

∆s→0

1∆

s”nˆ IC B · dl#

max

.

• The normal vector, nˆ, of the area, ∆s, is oriented such

that the circulation is a maximum.

• If ∇ × B = 0, then B is said to be a conservative field.

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

8/14

Curl – Coordinate Systems

• Cartesian coordinates:

∇ × B = ∂Bz

∂y –

∂B

y

∂z !xˆ + ∂B ∂zx – ∂B ∂xz !yˆ + ∂B ∂xy – ∂B ∂yx !zˆ

• Cylindrical coordinates:

∇×B = 1

r

∂Bz

∂φ –

∂Bφ

∂z !ˆr+ ∂B ∂zr – ∂B ∂rz !φˆ+ 1r ∂r ∂ (rBφ)- ∂B ∂φr !zˆ

• Spherical coordinates:

∇ × B = 1

Rsinθ

∂

∂θ (Bφsinθ) – ∂B ∂φθ !Rˆ + R1 sin1 θ ∂B ∂φR – ∂R ∂ (RBφ)!θˆ

+

1 R

∂

∂R(RBθ) – ∂B ∂θR !φˆ

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

9/14

Vector Identities Involving the Curl

For any two vectors A and B,

• ∇ × (A + B) = ∇ × A + ∇ × B.

• ∇ · (∇ × A) = 0 for any vector, A.

• ∇ × (∇V ) = 0 for any scalar function, V .

These identities will be revisited in your future studies.

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

10/14

Stoke’s Theorem

• Stoke’s theorem converts the surface integral of the curl of

a vector over an open surface, S, into a line integral of the

vector along the contour, C, bounding the surface, S.

• This can be described mathematically as

ZS(∇ × B) · ds = IC B · dl.

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

11/14

Stoke’s Theorem – Verification

• If B = cosφ

r

zˆ, verify Stoke’s theorem over the above

cylindrical surface; r = 2, π/3 ≤ φ ≤ π/2, 0 ≤ z ≤ 3.

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

12/14

The Laplacian Operator

• The last topic of these mathematics lectures is a concept

you will use in your future studies; the Laplacian operator.

• On Cartesian coordinates, the Laplacian operator is as

follows.

∇2 = ∂2

∂x2 +

∂2

∂y2 +

∂2

∂z2 .

• For a vector, E = Exxˆ + Eyyˆ + Ezzˆ,

∇2E = ∇2Exxˆ + ∇2Eyyˆ + ∇2Ezzˆ.

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

13/14

Laplacian – Coordinate Systems

For some vector, V ,

• Cartesian coordinates:

∇2V = ∂2V

∂x2 +

∂2V

∂y2 +

∂2V

∂z2 .

• Cylindrical coordinates:

∇2V = 1

r

∂

∂r r

∂V

∂r ! + r12 ∂∂φ 2V2 + ∂∂z 2V2 .

• Spherical coordinates:

∇2V = 1

R2

∂

∂R R2 ∂V ∂R!+ R2sinθ 1 ∂θ ∂ sinθ ∂V ∂θ !+ R2sin 1 2θ ∂∂φ 2V2 .

Circulation &

Curl

EGB241

Circulation

Curl

Stoke’s

Theorem

Laplacian

Operator

Next Week

14/14

Next Week

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