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,
- Cartesian, or rectangular, coordinates
- Cylindrical, or circular, coordinates
- Spherical, or polar, coordinates.
Cartesian or Rectangular Coordinates (x, y, z)
A point P in Cartesian coordinates is represented as P(x, y, z).The ranges of coordinate variables are,
From the Figure b, it is understood that any point in a rectangular coordinate system is the intersection of three planes:
- constant x-plane,
- constant y-plane, and
- constant z-plane, which are mutually perpendicular.
(a) Cartesian coordinates
(b) Constant x, y, z planes
A vector in the Cartesian coordinate system is written as,
From the definitions of dot product, we see that,
From the definitions of cross product, we see that,
Cylindrical, or Circular, Coordinates (r, φ, z)
A point P in cylindrical coordinates is represented as P(r, φ, z).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,
From the Figure. (b), it is understood that any point in cylindrical coordinates is an intersection of three planes, viz.
- constant ‘r’ plane (a circular cylinder),
- constant φ plane (semi-infinite plane with its edge along the z-axis) and
- constant z-plane (parallel to xy-plane).
(a) Cylindrical coordinates
(b) Constant r, φ, z planes
A vector in cylindrical the coordinate system is written as,
where are the unit vectors along the r, φ and z directions, respectively.
From the definitions of dot product, we see that,
From the definitions of cross product, we see that,
Relations between Cartesian (x, y, z) and Cylindrical (r, φ, z) Coordinates
and
The relationships between the unit vectors are obtained from Fig. and are given as,
-----(i)
and
------(ii)
Unit-vector tranformation between Cartesian and cylindrical coordinates
Unit-vector tranformation between Cartesian and cylindrical coordinates
The relationships between the component vectors and are obtained by using Eq. (i) and Eq. (ii) and then rearranging the terms. This is given as,
Thus, the relationships between the component vectors can be written in matrix form as,
and
Spherical, or Polar, Coordinates (ρ, θ, φ)
A point P in spherical coordinates is represented as P(ρ, θ, φ).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) |
From Fig. (b), it is understood that any point in spherical coordinates is an intersection of three planes, viz,
- constant ‘ρ’ plane (a sphere with its centre at the origin),
- constant θ-plane (circular cone with z-axis as its axis and the origin at its vertex), and
- constant φ-plane (semi-infinite plane as in cylindrical coordinates).
A vector in the spherical coordinate system is written as,
where, are the unit vectors along the ρ, θ and φ directions, respectively.
From the definitions of dot product, we see that,
From the definitions of cross product, we see that,
(a) Spherical coordinates
(b) Constant ρ, θ, φ planes
(c) Point P and unit vectors in spherical coordinates
- Relations between Cartesian (x, y, z) and Spherical (ρ, θ, φ) Coordinates
The relationships between Cartesian (x, y, z) and spherical (ρ, θ, φ) coordinates can also be obtained from Fig. (a) and can be written as,
and
The relationships between the unit vectors are obtained from Fig. and are given as,
-------(i)
and
-------(ii)
Unit-vector transformation for Cartesian and spherical coordinates
The relationships between the component vectors and can be obtained by using Eq. (i) and Eq. (ii) and then rearranging the terms. This is written in matrix form as,
and
------(iv)
- Relations between Cylindrical (r, φ, z) and Spherical (ρ, θ, φ) Coordinates
The relationships between cylindrical (r, φ, z) and spherical (ρ, θ, φ) coordinates are obtained from Fig. (a) and are written as,
------(v)
and
------(vi)
The relationships between the unit vectors are obtained from the Figure and are given as,
(vii) |
(viii) |
Unit vector transformation for cylindrical and spherical coordinates
The relationships between the component vectors and can be obtained by using Eq. (vii) and Eq. (viii) and then rearranging the terms. This is written in matrix form as,
(ix) |
(x) |