Spherical Coordinate System is a type of orthogonal system which is frequently used in Electromagnetics problems.
In Electromagnetics, we study phenomena related to Electric field, Magnetic field, their interaction etc. Most of the quantities in Electromagnetics are time-varying as well as spatial functions. In order to describe spatial variations (variation in space) of these quantities, one must be able to define all points and vectors uniquely in space using a proper coordinate system.
Thre are different types of orthogonal coordinate systems- Cartesian (or rectangular), circular cylindrical, spherical, elliptic cylindrical, parabolic cylindrical, conical, prolate spheroidal, oblate spheroidal and ellipsoidal.
Spherical Coordinate System
In the Spherical Coordinate System, a hypothetical sphere is assumed to be passing through the required point and any point of the space is represented using three coordinates that are r, θ, and φ i.e. P (r, θ, φ).
r is the radius of the hypothetical sphere passing through the required point or the minimum distance of the point from the origin.
φ is called as the azimuthal angle which is angle made by the half-plane containing the required point with the positive X-axis. The anticlockwise direction of rotation i.e. from the +X axis to +Y axis is considered as a positive angle. It is the same as we have seen in the cylindrical coordinate system.
θ also called as co-latitude. It is the angle between the z-axis and the position vector of the point.
As r is the radius of the hypothetical sphere that is assumed to be passing through the required point; it can take any value from 0 to ∞.
As stated in cylindrical coordinates, φ may take any value between o to 2Π to cover complete space.
Constant Coordinate Surfaces and Lines
As seen in Cartesian and Cylindrical, if only one coordinate of three is given to be constant and rest two are variables then the required locus comes out to be a plane or surface. In other words, r = constant (φ, θ variables), φ = constant (r, θ variables) or θ = constant (r, φ variables) represents the equations of planes each.
If two coordinates of three are given to be constant and remaining one is variable then the required locus becomes the straight line. For example, the graph for r = constant, Φ = constant and θ is variable (not given) is a semicircular line along θ. Likewise, r = constant and θ = constant gives us a circular line along φ which is formed by the intersection of these two planes. Finally, θ = constant and φ = constant leads to a line along r.
Finally, if all three coordinates are given or constants, then we get a specific point in the space.
Conversion from Spherical Coordinate System to Cartesian Coordinate System
To minimize the computational labour while solving the electromagnetic problems, it is often needed to convert one coordinate system to other and vice versa.
Let the point P is represented as P (x, y, z) in the Cartesian system and as P (r, θ, φ) in Cylindrical System, then relation among the variables is given as-
The transformations of the vector from one to other (Cartesian system and Spherical system then) is given as-