Circulation & Curl

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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
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