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ELE 216
Electromagnetic Field
Dr. Maha Raof
Coordinate Systems
Coordinate Systems
➢ To describe a vector accurately, some specific lengths, directions, angles,
projections, or components must be given.
Types of Coordinate Systems
Rectangular Cylindrical Spherical
Coordinate Coordinates Coordinates
System System System
(x, y, z) (ρ, φ, z) (r, θ, φ)
Rectangular (Cartesian) Coordinate System
Coordinates axes Representing a point
Vector and Unit Vector in Cartesian Coordinate
Base Vectors
y1
z1
x1
ഥ𝒙 + 𝒚𝟏 𝒂
𝒓𝑶𝑷 = 𝒙𝟏 𝒂 ഥ𝒚 + 𝒛𝟏 𝒂
ഥ𝒛
𝒓𝑶𝑷 = 𝒙𝟐𝟏 + 𝒚𝟐𝟏 + 𝒛𝟐𝟏
Vector and Unit Vector in Cartesian Coordinate
▪ rp = ax + 2ay + 3az
▪ rQ = 2ax – 2ay + az
Vector and Unit Vector in Cartesian Coordinate
Vector Expressions in Rectangular Coordinates
➢ General Vector, B: B = |B| aB = Bx ax + By ay + Bz az
➢ Magnitude of B: |B| = 𝐵𝒙𝟐 + 𝐵𝑦2 + 𝐵𝑧𝟐
➢ Unit Vector in the Direction of B:
𝑩 𝑩
𝒂𝐵 = =
𝑩
𝐵𝑥2 + 𝐵𝑦2 + 𝐵𝑧𝟐
Differential Elements in Cartesian Coordinate
P point coordinates:
(x, y, z)
P` point coordinates:
x + dx, y + dy, z + dz
▪ Differential lengths
𝑑𝑙 = 𝑑𝑥 𝑎ො𝑥 𝑑𝑙 = 𝑑𝑦 𝑎ො𝑦 𝑑𝑙 = 𝑑𝑧 𝑎ො𝑧
▪ Differential areas
𝑑𝑆 = 𝑑𝑥𝑑𝑦 𝑎ො𝑧 𝑑𝑆 = 𝑑𝑥𝑑𝑧 𝑎ො𝑦
𝑑𝑆 = 𝑑𝑦𝑑𝑧 𝑎ො𝑥
▪ Differential volume dv = dxdydz
Vector and Unit Vector in Cartesian Coordinate
9
❑ Example:
Given three points in Cartesian co-ordinates system as A(3,-2,1), B(-3,-3,5), C (2, 6, -4).
Find: i) The vector from A to C ii) The unit vector from B to A
iii) The vector from A to the midpoint of the straight line joining B to C
Sol:
ഥ = 3ത
𝐀 𝐚𝑦 + 𝐚ത 𝑧 ,
𝐚𝑥 − 2ത ഥ = −3ത
𝐁 𝐚𝑥 − 3ത
𝐚𝑦 + 5ത
𝐚𝑧 , 𝐂ത = 2ത
𝐚𝑥 + 6ത
𝐚𝑦 − 4ത
𝐚𝑧
i)
Vector and Unit Vector in Cartesian Coordinate
10
❑ Example:
Given three points in Cartesian co-ordinates system as A(3,-2,1), B(-3,-3,5), C (2, 6, -4).
Find: i) The vector from A to C ii) The unit vector from B to A
iii) The vector from A to the midpoint of the straight line joining B to C
Sol:
ii)
Vector and Unit Vector in Cartesian Coordinate
11
❑ Example:
Given three points in Cartesian co-ordinates system as A(3,-2,1), B(-3,-3,5), C (2, 6, -4).
Find: i) The vector from A to C ii) The unit vector from B to A
iii) The vector from A to the midpoint of the straight line joining B to C
Sol:
iii)
Mid point
The vector from A to mid-point
Rectangular Coordinates
Cylindrical Coordinate
Circular Cylindrical Coordinates
Base Vectors
𝟎 ≤ 𝒓≤ ∞ 𝟎 ≤ ∅ ≤ 𝟐𝝅 −∞ ≤ 𝒛 ≤ ∞
Circular Cylindrical Coordinates
Circular Cylindrical Coordinates
𝑃 = 𝜌𝑎ത𝜌 + ∅𝑎ത∅ + 𝑧𝑎ത𝑧
Differential elements in Cylindrical Coordinates
Differential lengths
𝑑𝑙 = 𝑑𝜌 𝑎ො𝜌
𝑑𝑙 = 𝜌𝑑∅ 𝑎ො∅
𝑑𝑙 = 𝑑𝑧 𝑎ො𝑧
Differential elements in Cylindrical Coordinates
ෝ𝒛
𝒂
ෝ𝒓
𝒂
ෝ∅
𝒂
• Differential volume
Spherical Coordinates systems
𝒓≥𝟎 𝟎 ≤ ∅ ≤ 𝟐𝝅 𝟎 ≤ 𝜽 ≤ 𝝅
Point P specified by P(r, 𝜽, ∅)
Spherical Coordinates systems
𝒓≥𝟎 𝟎 ≤ ∅ ≤ 𝟐𝝅 𝟎 ≤ 𝜽 ≤ 𝝅
Point P specified by P(r, 𝜽, ∅)
Spherical Coordinates systems
Spherical Coordinates systems
Base Vectors
𝒓≥𝟎 𝟎 ≤ ∅ ≤ 𝟐𝝅 𝟎 ≤ 𝜽 ≤ 𝝅
Differential elements Coordinates systems
▪ Differential lengths
𝑑𝑙 = 𝑑𝑟 𝑎ො𝑟 𝑑𝑙 = 𝑟𝑑𝜃 𝑎ො𝜃 𝑑𝑙 = 𝑟 sin 𝜃 𝑑∅ 𝑎ො∅
▪ Differential areas
𝑑𝑆 = 𝑟 2 sin 𝜃 𝑑𝜃𝑑∅ 𝑎ො𝑟
𝑑𝑆 = 𝑟 sin 𝜃 𝑑𝑟𝑑∅ 𝑎ො𝜃
𝑑𝑆 = 𝑟𝑑𝑟𝑑𝜃 𝑎ො∅
▪ Differential volume
𝑑𝑉 = 𝑟 2 sin 𝜃 𝑑𝑟𝑑𝜃𝑑∅
Spherical Coordinates systems
Vector Transformation
Cylindrical Cartesian
Cylindrical and Cartesian Coordinate Systems
➢ Transform the vector: into cylindrical coordinates
Sol:
𝐵𝜌 = 𝐁 . 𝐚𝛒 = 𝑦 𝐚𝐱 . 𝐚𝛒 − 𝑥 𝐚𝒚 . 𝐚𝛒 + 𝑧(𝐚𝐱 . 𝐚𝛒 )
𝐵∅ = 𝐁 . 𝐚∅ = 𝑦 𝐚𝐱 . 𝐚∅ − 𝑥 𝐚𝒚 . 𝐚∅ + 𝑧(𝐚𝐱 . 𝐚∅ )
𝐵𝑧 = 𝐁 . 𝐚𝐳 = 𝑦 𝐚𝐱 . 𝐚𝒛 − 𝑥 𝐚𝒚 . 𝐚𝒛 + 𝑧(𝐚𝒛 . 𝐚𝒛 )
Relation between Cartesian and Cylindrical systems
Relation between Cartesian and Cylindrical systems
𝝆 −𝑎ො𝑥 𝑎ො∅
∅
∅ 𝑎ො𝑦
90𝑜 − ∅
𝑎ො𝑥 ∅
𝑎ො𝜌
Relation between Cartesian and Cylindrical systems
Cylindrical and Cartesian Coordinate Systems
➢ Transform the vector: into cylindrical coordinates
Sol: 𝐵𝜌 = 𝐁 . 𝐚𝛒 = 𝑦 𝐚𝐱 . 𝐚𝛒 − 𝑥 𝐚𝒚 . 𝐚𝛒 + 𝑧 𝐚𝒛 . 𝐚𝛒
= y cos∅ − x sin∅ = 𝜌 sin∅ cos∅ − 𝜌 cos∅ sin∅ = 0
𝐵∅ = 𝐁 . 𝐚∅ = 𝑦 𝐚𝐱 . 𝐚∅ − 𝑥 𝐚𝒚 . 𝐚∅ + 𝑧 𝐚𝒛 . 𝐚∅
= −y sin∅ − x cos∅ = 𝜌 𝑠𝑖𝑛2 ∅ − 𝜌 𝑐𝑜𝑠 2 ∅ = − 𝜌
𝐵𝑧 = 𝐁 . 𝐚𝐳 = 𝑦 𝐚𝐱 . 𝐚𝒛 − 𝑥 𝐚𝒚 . 𝐚𝒛 + 𝑧 𝐚𝒛 . 𝐚𝒛 = 𝑧
Spherical Cartesian
Relation between Cartesian and Spherical systems
Spherical and Rectangular Coordinate Systems
𝐚ො 𝐫
𝟗𝟎 − ∅ 𝐚
ො∅
∅
∅
𝐚ො 𝐲
𝐚ො 𝐱
Spherical and Rectangular Coordinate Systems
Spherical and Rectangular Coordinate Systems
Example
Transform the vector: into spherical coordinates
Vector Calculus
Divergence of a vector field
Divergence of a vector field
Divergence of a vector field
Divergence is the outflow of flux from a small closed surface per unit
volume as the volume shrinks to zero.
➢ Mathematical definition:
A
Divergence of a vector field
➢ Divergence in different coordinate systems: 𝛁. 𝐃
Type
equation
here.
Curl of a vector field
Curl is a measure of how much a vector field circulates or rotates
about a given point
➢Mathematical definition:
Curl of a vector field
Curl of a vector field
Rectangular coordinates
Cylindrical coordinates
Spherical coordinates
Thank You
