Differential Analysis - Cylindrical Coordinate System - Field Theory.
Differential Length(dl):
In General the differential length is given as
dl = dρ aρ + ρdφ aφ+ dz az
Differential Length for a surface is given as:
- dl = dρ aρ+ ρdφ aφ ---(For ρ-φ Plane or Z constant Plane)
- dl = ρdΦ aφ+ dz az ---(For φ-z Plane or ρ Constant Plane)
- dl = dρ aρ+ dz az ---(For ρ-z Plane or φ Constant Plane)
Differential length for a line parallel to ρ, φ and z axis are respectively given as:
- dl = dρ aρ ---(For a line parallel to ρ axis)
- dl = ρdφ aφ ---(For a line parallel to φ axis)
∫ dl = ∫o2π ρdφ = ρ( 2π - 1) = 2πρ
This resembles the circumference of a circle. Hence if φ varies with ρ and z constant, then the length is the circumference of the circle.
dl = dzaz ---(For a line parallel to z axis)
Differential Surface (ds):
- ds = ρdρ dφ az
This surface describes a circular disc. Always remember- To define a circular disk we need two parameter one distance measure and one angular measure. An angular parameter will always give a curved line or an arc.
In this case dΦ is measured in terms of change in arc.
Arc is given as:
Arc= radius * angle
Therefore, whenever there is a change in angle the radius always remains constant. Hence ρ always assist dφ.
∫ ds = ∫oρ∫o2π ρ dρ dφ = (ρ2/2) (2π -1) = π ρ2
This answer describes the surface area of a circle. Hence the surface is a circular disc.
- ds = ρdφ dz aρ
This surface describes the curved surface of the cylinder. We can also say that this surface defines a hollow cylinder.
Suppose the height of the cylinder varies from 0 to h.
∫ ds = ∫oh∫o2π ρ dφ dz = ρ (h - 0) (2π - 0) = 2 πρh
This answer describes the surface area of a cylinder. Hence the surface is a hollow cylinder.
- ds = dρ dz aφ
This surface describes a simple ρ-z plane which is along the direction of φ.
Differential Volume (dv):
dv = ρdρ dφ dz ---(Scalar Quantity)
∫ dv = ∫oρ ∫o2π ∫oh ρ dρ dφ dz = (ρ2/2) (2π -0) (h -0) = π ρ2 h
This answer describes the volume of a cylinder.
In General the differential length is given as
dl = dρ aρ + ρdφ aφ+ dz az
Differential Length for a surface is given as:
- dl = dρ aρ+ ρdφ aφ ---(For ρ-φ Plane or Z constant Plane)
- dl = ρdΦ aφ+ dz az ---(For φ-z Plane or ρ Constant Plane)
- dl = dρ aρ+ dz az ---(For ρ-z Plane or φ Constant Plane)
Differential length for a line parallel to ρ, φ and z axis are respectively given as:
- dl = dρ aρ ---(For a line parallel to ρ axis)
- dl = ρdφ aφ ---(For a line parallel to φ axis)
∫ dl = ∫o2π ρdφ = ρ( 2π - 1) = 2πρ
This resembles the circumference of a circle. Hence if φ varies with ρ and z constant, then the length is the circumference of the circle.
dl = dzaz ---(For a line parallel to z axis)
Differential Surface (ds):
- ds = ρdρ dφ az
This surface describes a circular disc. Always remember- To define a circular disk we need two parameter one distance measure and one angular measure. An angular parameter will always give a curved line or an arc.
In this case dΦ is measured in terms of change in arc.
Arc is given as:
Arc= radius * angle
Therefore, whenever there is a change in angle the radius always remains constant. Hence ρ always assist dφ.
∫ ds = ∫oρ∫o2π ρ dρ dφ = (ρ2/2) (2π -1) = π ρ2
This answer describes the surface area of a circle. Hence the surface is a circular disc.
- ds = ρdφ dz aρ
This surface describes the curved surface of the cylinder. We can also say that this surface defines a hollow cylinder.
Suppose the height of the cylinder varies from 0 to h.
∫ ds = ∫oh∫o2π ρ dφ dz = ρ (h - 0) (2π - 0) = 2 πρh
This answer describes the surface area of a cylinder. Hence the surface is a hollow cylinder.
- ds = dρ dz aφ
This surface describes a simple ρ-z plane which is along the direction of φ.
Differential Volume (dv):
dv = ρdρ dφ dz ---(Scalar Quantity)
∫ dv = ∫oρ ∫o2π ∫oh ρ dρ dφ dz = (ρ2/2) (2π -0) (h -0) = π ρ2 h
This answer describes the volume of a cylinder.
SOLVED EXAMPLE / NUMERICALS:
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) ax + xyz ay + (x2 – z2) ax to cylindrical and spherical system? SOLUTION/ANSWER
Q.5 Express the vector A = ρ (z2 + 1)aρ - ρz cosφaφ in Cartesian co-ordinate system? SOLUTION/ANSWER
Q.6 Express the vector E =2r sinθ cosφar + r cosθ cosφaθ – r sinφaφ 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.
- Circular Cylindrical Coordinate System (ρ, φ, z).
- 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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