Cylindrical Coordinate System

Cylindrical Coordinate System is a type of orthogonal system which is frequently used in Electromagnetics problems involving circular fields or forces.

Electromagnetism is a branch of Physics which deals with the study of phenomena related to Electric field, Magnetic field, their interactions etc. Most of the quantities in Electromagnetics are functions of time as well as space.

Spatial variations i.e. variations in space requires unique representation of a point of interest using suitable reference. Such reference systems are called as coordinate systems. There are many types of coordinate systems as a Cartesian coordinate system, circular cylindrical, spherical, elliptic cylindrical, parabolic cylindrical, conical, prolate spheroidal, oblate spheroidal and ellipsoidal.

In the Cylindrical Coordinate System, any point of the space is represented using three coordinates that are ρ, φ and z. Any point in this system is represented as P (ρ, φ, z).

ρ is the radius of the cylinder passing through P or the radial distance from the z-axis.
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.
z coordinate is the same as in the Cartesian system; it is the distance of the required point from the XY plane.

As ρ is the radius of the hypothetical cylinder that is assumed to be passing through the required point; it can take any value from 0 to ∞.
φ is the angle as shown in the figure and to cover complete space it may take any value between o to 2Π.
z is the distance of the required point from XY plane and it is exactly the same as in Cartesian Coordinate System. To cover complete space its value can vary in between -∞ to +∞.

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Constant Coordinate Surfaces and Lines

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, ρ = constant (φ, z variables), φ = constant (ρ, z variables) or z = constant (ρ, 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 ρ = constant, φ = constant and z is variable (not given) is a straight line parallel to Z-axis. Likewise, ρ = constant and z = constant gives us a circular line i.e. along φ and φ = constant and z = 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.

Vectors in Cylindrical Coordinate System

Any vector in a Cylindrical coordinate system is represented using three mutually perpendicular unit vectors.

The general form of any vector in the Cylindrical coordinate system is as follows-

A_ρ is the ρ-component, A_φ is the φ-component and A_z is the z-component of given vector. Component of the vector is a flux or effect of the given vector along the required axis or direction.

Conversion from Cylindrical Coordinate System to Cartesian Coordinate System

While solving Electromagnetic problems, many times is required to convert one coordinate system to other. The selection of the coordinate system should be in accordance with the given problem to minimize the calculation labor.

Let the point P is represented as P (x, y, z) in the Cartesian system and as P (ρ, φ, z) in Cylindrical System, then relation among the variables is given as-