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Coordinate Systems
1 Introduction
Electromagnetics is the study of the effects of electric charges in rest and motion. Some fun- damental quantities in electromagnetics are scalars while others are vectors. These scalars and vectors can be functions of position and time and they can be completely described using an appropriate coordinate system. The laws of electromagnetism are invariant with coordinate sys- tems i.e., they can be completely described using an appropriate coordinate system and must hold good irrespective of the coordinate system used. The choice of a specific coordinate system is decided by the geometry of the given problem. There are 8 orthogonal coordinate systems, namely
- Cartesian Coordinate System
- Cylindrical Coordinate System
- Spherical Coordinate System
- Parabolic Cylindrical Coordinate System
- Conical Coordinate System
- Prolate Spheroidal Coordinate System
- Oblate Spheroidal Coordinate System
- Ellipsoidal Coordinate System
Of these 8 orthogonal coordinate systems, the most widely used are the Cartesian, cylindrical and spherical coordinate system. We will be using these three systems in all of our discussions.
2 Coordinate Basis
A point in a three dimensional space can be located as the intersection of three surfaces. For ex- ample, the corner of a room is the point of intersection of the three walls representing the three planes. When these three planes are orthogonal to each other, we have an orthogonal coordinate system. The three orthogonal surfaces can be represented as ui = constant (i = 1, 2 , 3). These surfaces need not be planes; they could be curved surfaces. The unit vector in the three coordi- nate directions are called base vectors. Any point in space can be written as a linear combination of these base vectors. We now describe the base vectors of each coordinate system. Usually right handed coordinate systems are chosen to describe the problems in electromagnetics.
2.1 Cartesian System
In the Cartesian system, the 3 base vectors are ˆax, ˆay and ˆaz. Any point in space can be written in the form,
−→ P = x 1 aˆx + y 1 ˆay + z 1 ˆaz
where (x 1 , y 1 , z 1 ) are the coordinates of the point P in the Cartesian space which is the inter- section of the three planes x = x 1 , y = y 1 , z = z 1. The distance of the point from the origin is given by,
P | =
x^21 + y 12 + z^21
The figure below depicts the point P in the Cartesian Coordinate System. As you can see x 1 ,y 1 and z 1 can also be understood as the perpendicular distance of point P from the YZ, XZ and XY plane.
Figure 1: Cartesian coordinate system
Figure 2: Differential elements in Cartesian system
The differential length, differential surface and differential volume are given by
From fig.4 we can see that the differential length, differential surface and differential volume are given by
−→ dl = drˆar + rdφˆaφ + dzˆaz −→ dSr = rdφdzˆar −→ dSφ = drdzˆaφ −→ dSz = rdφdrˆaz dV = rdrdφdz
Figure 4: Differential elements in cylindrical system
2.3 Spherical System
In the Spherical system, the 3 basis are ˆar, ˆaθ and ˆaφ. Any point in space can be written in the form,
−→ P = r 1 ˆar + θ 1 ˆaθ + φ 1 ˆaφ
where (r, θ, φ) are the coordinates of the point P in the Spherical Space. A point P(r 1 , θ 1 , φ 1 ) in the spherical coordinates is specified as the intersection of the following three surfaces: a spherical surface centered at origin and has a radius r 1 , a right circular cone with its apex at origin and half angle θ 1 and a half plane containing z-axis and making an angle φ 1 with the XZ plane.
Figure 5: Spherical coordinate system
The distance of the point from the origin is given by,
|
P | = r 1
The values of the 3 coordinates vary as follows,
r [0, ∞) θ [0, π] φ [0, 2 π)
Figure 6: Differential coordinates in spherical system
From fig.6 we can see that the differential length, differential surface and differential volume are given by
−→ dl = drˆar + rdθˆaθ + r sin θdφˆaφ −→ dSr = r^2 sin θdθdφˆar −→ dSθ = r sin θdrdφˆaθ −→ dSφ = rdrdθaˆφ dV = r^2 sin θdrdθdφ
3 Properties of Basis Vectors
3.1 Cartesian System
ax.ax = ay.ay = az .az = 1 ax.ay = ay.az = az .ax = 0 ax × ay = az , ay × az = ax , az × ax = ay
3.2 Cylindrical System
ar.ar = aφ.aφ = az .az = 1 ar.aφ = aφ.az = az .ar = 0 ar × aφ = az , aφ × az = ar , az × ar = aφ
- Vector functions - Those functions whose values are vectors −→v = −→v (−→ P ) = [v 1 (−→ P ), v 2 (−→ P ), v 3 (−→ P )]
where v 1 , v 2 , v 3 are the coordinates representing the direction of the vector at the given point
P. Eg. Electric field due to a point charge, normal fields of a surface, tangential fields to a curve etc.
Figure 8: Normals fields to a surface and tangential fields to a curve
Vector and scalar functions may depend on time in addition to their dependencies on space.
5.2 Integrals
There are 3 types of integrals in vector calculus. They are
- Line Integral - It is the integral of the tangential component of a vector field
A along a curve L. It is given by
L
A.
dl. Fig. 7 shows the line integral of a vector field (blue) over a closed path (red).
Figure 9: Line integral
- Surface Integral - Given a vector field
A continuous in a smooth surface S, the surface integral is the flux of
A through S. It is given by
S
A.
dS. Fig. 10 shows the surface integral of a vector field (red) over a surface (blue).
Figure 10: Surface integral
- Volume Integral - Volume integral of a scalar ρv over the volume V is given by
V ρvdV^. dV is the differential volume.
The Del Operator (∇) plays a very important role in vector calculus. It is given by −→ ∇ =
∂x
ˆax +
∂y
ˆay +
∂z
ˆaz
∇ is used to define gradient and laplacian of a scalar field and divergence and curl of a vector field.
5.3 Gradient of a Scalar Field
Gradient of a scalar field V, is a vector that represents both magnitude and direction of max- imum space rate of change of V. The gradient of V, ∇V , will always be perpendicular to a constant V surface. If
A = ∇V , then V is said to be the scalar potential of
A. Consider a scalar function - say temperature distribution inside a room. The magnitude of this function depends on the position. Consider two surfaces on which the magnitude of V is a constant with values V 1 and V 1 + dV respectively, where dV indicates a small change in V.
Figure 11: Gradient of a scalar function
Given below is the divergence operator in the 3 coordinate systems.
In Fig. 12.(a) and 12.(b) there is no divergence and in Fig.12(c) there is a positive divergence. A positive divergence would imply the presence of a source in the given volume and negative divergence would imply a sink. This source is also referred to as the ’flow source’ and div
A is a measure of the strength of flow source. Divergence theorem relates the divergence of a vector field
A to the surface integral of −→ A over a surface. It is given by ∮
S
A.
dS =
V
A dV
where S is the surface and V in the volume enclosed by the surface S.
Figure 13: Divergence of a field inside the spherical surface is equal to the surface integral of the field on the surface
This applies to any surface volume V that is bounded by the surface S. Direction of d
S is always that of the outward normal perpendicular to the surface d
S and directed away from the volume.
5.4.1 Curl
Similar to a flow source, vector fields can also exist as ’vortex sources’ which causes circulation if a vector field around it. Net circulation of a vector field around a closed path is defined as,
C
A .d
l , the closed line
integral over the path. If
A is a force acting on an object, circulation would be the work done by the force in moving th object once around the contour.
The strength of the vortex source can be defined through curl of the vector
A. Circulation is a line integral of a dot product, whose value depends on the orientation of the contour C, with respect to the vector
A. To define point function, the contour C is shrunk and oriented such that the circulation is maximum. Curl of
A is a vector whose magnitude is the maximum circulation of
A per unit area, as the area tends to zero. Direction of the curl is the normal direction of the area when the area is oriented to make the circulation maximum.
curl
A =
∇ ×
A = lim ∆S→ 0
[ ∮
c
A.
dl ∆S aˆn
]
max Stokes Theorem relates the curl of a vector field
A to the line integral of
A over a contour C. It is given by ∮
C
A.
dl =
S
∇ ×
A ).
dS
The surface integral of a vector field over an open surface is equal to the closed line integral of the vector along the contour bounding the surface.
Figure 14: Surface integral of the field on the surface is equal to the line integral of the field along the contour enclosing the surface
Given below is the ∇ operator in the 3 coordinate systems.
