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Coordinate Systems

In electromagnetics, most of the quantities are functions of space and time. In order to describe the spatial variations of these quantities, all the points in space must be defined uniquely using an appropriate coordinate system.

We will discuss the most useful three coordinate systems, namely,
  1. Cartesian, or rectangular, coordinates
  2. Cylindrical, or circular, coordinates
  3. Spherical, or polar, coordinates.

Cartesian or Rectangular Coordinates (xyz)

A point P in Cartesian coordinates is represented as P(xyz).

The ranges of coordinate variables are,
 
Description: 27430.png

From the Figure b, it is understood that any point in a rectangular coordinate system is the intersection of three planes:
  1. constant x-plane,
  2. constant y-plane, and
  3. constant z-plane, which are mutually perpendicular.
Description: 97516.png
(a) Cartesian coordinates

Description: 97549.png
(b) Constant xyz planes
 
A vector Description: 27447.png in the Cartesian coordinate system is written as,
 
Description: 27457.png
 
where, Description: 27466.png are the unit vectors along the xy and z directions, respectively.

From the definitions of dot product, we see that,
 
Description: 27474.png
From the definitions of cross product, we see that,
 
Description: 27489.png

Cylindrical, or Circular, Coordinates (rφz)

A point P in cylindrical coordinates is represented as P(rφz).
 
Here,
r = Radius of the cylinder passing through P = Radial distance from the z-axis
φ = Angle measured from the x-axis in the xy-plane, known as azimuthal angle
z = Same as in Cartesian coordinates

The ranges of coordinate variables are,
 
Description: 27509.png
From the Figure. (b), it is understood that any point in cylindrical coordinates is an intersection of three planes, viz.
  1. constant ‘r’ plane (a circular cylinder),
  2. constant φ plane (semi-infinite plane with its edge along the z-axis) and
  3. constant z-plane (parallel to xy-plane).
Description: 97684.png
(a) Cylindrical coordinates

Description: 97718.png
(b) Constant rφz planes
 
A vector Description: 27523.png in cylindrical the coordinate system is written as,
 
Description: 27532.png
 
where Description: 27543.png are the unit vectors along the rφ and z directions, respectively.

From the definitions of dot product, we see that,
 
Description: 27553.png
From the definitions of cross product, we see that,
 
Description: 27572.png

Relations between Cartesian (x, y, z) and Cylindrical (rφ, z) Coordinates
 
The relationships between Cartesian (xyz) and cylindrical (rφz) coordinates are obtained from
 
Fig. (a) and are written as,
 
Description: 27588.png
and
 
Description: 27667.png

The relationships between the unit vectors are obtained from Fig. and are given as,
 
Description: 27681.png-----(i)
and
Description: 27698.png------(ii)


Unit-vector tranformation between Cartesian and cylindrical coordinates
The relationships between the component vectors Description: 27719.png and Description: 27726.png are obtained by using Eq. (i) and Eq. (ii) and then rearranging the terms. This is given as,
 
Description: 27734.png
 
Thus, the relationships between the component vectors can be written in matrix form as,
 
Description: 27750.png and Description: 70091.png

Spherical, or Polar, Coordinates (ρ, θ, φ)

A point P in spherical coordinates is represented as P(ρθφ).
 
Here,
ρ = Distance of the point from the origin
  = Radius of a sphere centred at the origin and passing through the point P
θ = Angle between the z-axis and the position vector P, known as colatitudes, and
φ = Angle measured from the x-axis in the xy-plane, known as azimuthal angle (same as in cylindrical coordinates)
 
The ranges of coordinate variables are,
 
Description: 27773.png
From Fig. (b), it is understood that any point in spherical coordinates is an intersection of three planes, viz,
  1. constant ‘ρ’ plane (a sphere with its centre at the origin),
  2. constant θ-plane (circular cone with z-axis as its axis and the origin at its vertex), and
  3. constant φ-plane (semi-infinite plane as in cylindrical coordinates).
A vector Description: 27790.png in the spherical coordinate system is written as,
 
Description: 27801.png
where, Description: 27810.png are the unit vectors along the ρθ and φ directions, respectively.

From the definitions of dot product, we see that,
 
Description: 27817.png
From the definitions of cross product, we see that,
 
Description: 97815.png


(a) Spherical coordinates


(b) Constant ρθφ planes
 
Description: 58349.png 
(c) Point P and unit vectors in spherical coordinates
  1. Relations between Cartesian (x, y, z) and Spherical (ρθφ) Coordinates
The relationships between Cartesian (xyz) and spherical (ρθφ) coordinates can also be obtained from Fig. (a) and can be written as,
 
Description: 27851.png
and
 
Description: 27865.png
 
The relationships between the unit vectors are obtained from Fig. and are given as,
 
 
Description: 27882.png -------(i)
 
and
 
Description: 27902.png-------(ii)
 
Unit-vector transformation for Cartesian and spherical coordinates
 
The relationships between the component vectors Description: 27917.png and Description: 27924.png can be obtained by using Eq. (i) and Eq. (ii) and then rearranging the terms. This is written in matrix form as,
 
 
Description: 27934.png------(iii)
and
Description: 27950.png------(iv)
  1. Relations between Cylindrical (rφ, z) and Spherical (ρθφ) Coordinates
The relationships between cylindrical (rφz) and spherical (ρθφ) coordinates are obtained from Fig. (a) and are written as,
 
Description: 27965.png------(v)
 
and
 
Description: 27985.png------(vi)
 
The relationships between the unit vectors are obtained from the Figure and are given as,
 
Description: 28001.png (vii)
 
and
 
Description: 28017.png (viii)
 
Unit vector transformation for cylindrical and spherical coordinates

The relationships between the component vectors Description: 28035.png and Description: 28042.png can be obtained by using Eq. (vii) and Eq. (viii) and then rearranging the terms. This is written in matrix form as,
 
Description: 28049.png (ix)
 
and
 
Description: 28067.png (x)





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