Introduction Gradient of a scalar field Divergence of a vector field – Divergence Theorem Curl of a vector field – Stoke’s Theorem
Vector.2
Scalar and Vector Scalar – Can be completely specified by its magnitude – Can be a complex number – Examples: • Voltage: 2V, 2.5∠10° • Current • Impedance: 10+j20Ω
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Scalar and Vector Scalar field – A scalar which is a function of position – Example: T=10+x • Represented by brightness in this picture
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Scalar and Vector Vector – Specify both the magnitude and direction of a quantity – Examples • Velocity: 10m/s along x-axis • Electric field: y-directed electric field with magnitude 2V/m Vector field – Example
T = xˆ
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Addition Sum of two vectors
C= A+B =B+A Graphical representation
Example
A = 2 xˆ B = 0 . 7 xˆ + yˆ ∴ C = A + B = 2 . 7 xˆ + yˆ Vector.6
Scalar Multiplication Simple product – Multiplication of a scalar
C = aB – Direction does not change
aB B
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Scalar or Dot Product
A ⋅ B = AB cos θ AB
θ AB is the angle between the vectors. – The scalar product of two vectors yields a scalar whose magnitude is less than or equal to the products of the magnitude of the two vectors. – When the angle θ AB is 90°, the two vectors are orthogonal and the dot product of two orthogonal vectors is zero. – Example:
A = 10 xˆ + 2 yˆ B = 3 xˆ A ⋅ B = (10 xˆ + 2 yˆ ) ⋅ 3 xˆ = 30 xˆ ⋅ xˆ + 6 yˆ ⋅ xˆ = 30 Vector.8
Vector or Cross Product
A × B = nˆ AB sin θ AB – θ AB is the angle between the vectors – nˆ is a unit vector normal to the plane containing the vectors • Right-hand rule
nˆ
A × B = −B × A Vector.9
Vector or Cross Product In cartesian coordinate system,
xˆ × yˆ = zˆ yˆ × zˆ = xˆ zˆ × xˆ = yˆ xˆ
yˆ
zˆ
A × B = Ax Bx
Ay By
Az Bz
Timeout – M3.1 – 3.4 Vector.10
Orthogonal Coordinate Systems In electromagnetics, the fields are functions of space and time. A three-dimensional coordinate system allow us to uniquely specify the location of a point in space or the direction of a vector quantity. – Cartesian (rectangular) coordinate system – Cylindrical coordinate system – Spherical
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Cartesian Coordinates (x,y,z) Differential length: d l = xˆ dx + yˆ dy + zˆ dz Differential surface area:
Fig. 3-8
d s x = xˆ dydz d s y = yˆ dxdz d s z = zˆ dxdy Differential volume:
dv = dxdydz
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Cylindrical Coordinates
(r ,φ , z )
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Cylindrical Coordinates Differential length:
d l = rˆdr + φˆrd φ + zˆ dz
Differential surface area:
d s r = rˆrd φ dz d s = φˆdrdz φ
d s z = zˆ rd φ dr Differential volume:
dv = rdrd φ dz
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Example 3-4
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Spherical Coordinates
( R ,θ ,φ )
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Spherical Coordinates Differential length:
d l = Rˆ dR + θˆRd θ + φˆR sin θ d φ Differential surface area:
d s R = Rˆ R 2 sin θ d θ d φ d s = θˆR sin θ dRd φ θ
d s φ = φˆRdRd θ Differential volume:
dv = R 2 sin θ dRd θ d φ
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Example 3-5
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Summary
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Gradient of a Scalar Field In Cartesian coordinate, the gradient of scalar field T is
∂f ∂f ∂f grad f = ∇ f = yˆ + zˆ xˆ + ∂x ∂y ∂z – a vector in the direction of maximum increase of the field f. – ∇ is an operator and defined as
∂ ∂ ∂ ∇ ≡ xˆ + yˆ + zˆ ∂x ∂y ∂z
Demonstration: D3.1, D3.2, DM3.5, M3.6 Vector.20
Del Operator The operator in cylindrical coordinates is defined as
∂ 1 ∂ ˆ ∂ ∇ ≡ rˆ + φ + zˆ ∂r r ∂φ ∂z
In spherical coordinates, we have
∂ ˆ 1 ∂ ˆ 1 ∂ ˆ ∇ ≡ R+ θ + φ ∂R R ∂θ R sin θ ∂ φ
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Divergence of a Vector Field Divergence of a vector field A:
div A ≡ lim
∆v→ 0
∫ A ⋅ dS S
∆v
If we consider the vector field A as a flux density (per unit surface area), the closed surface integral represents the net flux leaving the volume ∆v In rectangular coordinates,
∂Ay ∂Ax ∂Az + + div A = ∇ ⋅ A = ∂x ∂y ∂z D3.10, M3.8 Vector.22
Divergence Theorem If A is a vector, then for a volume V surrounded by a closed surface S,
∫
V
∇ ⋅ A dv =
∫ A ⋅ dS S
The above integral represents the net flex leaving the closed surface S if A is the flux density V
S Vector.23
Curl of a Vector Field The curl of a vector field describes the rotational property, or the circulation of the vector field. Examples:
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Curl of a Vector Field
curl A ≡ ∇ × A ≡ lim
∆S → 0
nˆ ∫ A ⋅ d l C
∆S
In Cartesian coordinates, the curl of a vector is xˆ ∂ ∇×A = ∂x Ax
yˆ ∂ ∂y Ay
zˆ ∂ ∂z Az Vector.25
Stoke’s Theorem Stokes’s theorem: For an open surface S bounded by a contour C,
∫
S
(∇ × A ) ⋅ d S =
∫ A ⋅ dl C
C
S
The line integrals from adjacent cells cancel leaving the only the contribution along the contour C which bounds the surface S. Vector.26