Circular Cylindrical Coordinate System (ρ, φ, z) - Field Theory.

A Vector in Cylindrical system is represented as
(A_ρ, A_φ, A_z)
or
A = A_ρa_ρ+ A_φa_φ+ A_za_z

Where a_ρ, a_φand a_z are the unit vectors in ρ, φ and z direction respectively.


The physical significance of each parameter of cylindrical coordinates:

     - The value ρ indicates the distance of the point from the z-axis. It is the radius of the cylinder.

     - The value φ, also called the azimuthal angle, indicates the rotation angle around the z-axis. It is basically measured from the x axis in the x-y plane. It is measured anti-clockwise.


     - The value z indicates the distance of the point from z-axis. It is the same as in the Cartesian system. In short, it is the height of the cylinder.



Range of the variables:

It defines the minimum and the maximum value that ρ, φ and z can have in Cartesian system.

 0 ≤ ρ ≤ ∞
 0 ≤ φ ≤ 2π
-∞ ≤ z ≤ ∞

 Figure shows Point P and Unit vectors in Cylindrical Co-ordinate System.



Cylindrical System - Unit vectors:

Since the co-ordinate system is orthogonal, the unit vectors a_ρ, a_φ and a_z are mutually perpendicular to each other.

• a_ρ points in the direction of increasing ρ, i.e a_ρ points away from the z-axis.
• a_φ points in the direction of increasing φ (anticlockwise).
• a_z points in the direction of increasing z.

Relationship between Cylindrical and Cartesian Co-ordinate System Parameters:


Consider the parallelogram ABOC,

X = ρcosφ.
Y = ρsinφ.
Z = Z.

From the above equations we have,




Relationship between (a_x, a_y and a_z) and (a_ρ , a_φand a_z) :

Since a_z is common between the two coordinate system, our main focus is to find out the relation between a_x, a_y and a_ρ , a_φ

We know φ is the angle from the x-axis on the x-y plane.

From the above figures two equations can be deduced,


Transformation of vector A from (A_x, A_y, A_z) to (A_ρ, A_φ, A_z) i.e. transformation of Vector A from Cartesian to Cylindrical can be obtained as



Transformation of vector A from (A_ρ, A_φ, A_z) to (A_x, A_y, A_z) i.e. transformation of Vector A from Cylindrical to Cartesian can be obtained as






SOLVED EXAMPLE / NUMERICAL:

Q.2. Express the point P (1, -4, -3) in cylindrical and spherical co-ordinates? SOLUTION/ANSWER


Q.3
a) If V = XZ – XY +YZ, express V in Cylindrical co-ordinate system.

b) If U = X² + 2Y² +3z², express U in Spherical co- ordinates System.   SOLUTION/ANSWER


Q.4 Transform the vector E = (y² – x²) a_x + xyz a_y + (x² – z²) a_x to cylindrical and spherical system?      SOLUTION/ANSWER


Q.5 Express the vector A = ρ (z² + 1)a_ρ - ρz cosφa_φ in Cartesian co-ordinate system?       SOLUTION/ANSWER


Q.6 Express the vector E =2r sinθ cosφa_r + r cosθ cosφa_θ – r sinφa_φ in Cartesian co-ordinate system?SOLUTION/ANSWER


Q.7 Calculate the distance between the following pair of points?

a) (2, 1, 5) and (6, -1, 2)
b) (3, π/2, -1) and (5, 3π/2, 5)
c) (10, π/4, 3π/4) and (5, π/6, 7π/4)     SOLUTION/ANSWER


ALSO READ:

- Introduction To Coordinate System.

- Cartesian Coordinate System / Rectangular Coordinate System (x, y, z).

- Differential Analysis Of Cartesian Coordinate System.

- Differential Analysis Of Cylindrical Coordinate System.

- Spherical Coordinate System ( r, θ , φ).

- Differential Analysis Of Spherical Coordinate System.

- Numericals / Solved Examples - Page 1.

- Numericals / Solved Examples - Page 2.

- Short Notes/FAQ's

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