Differential analysis - Spherical Co-ordinate System - Field Theory.

Always remember that for a small change in angle the radius remains constant.

Radius of a cylinder (ρ) is constant for a small change in φ . Similarly the radius of a sphere(r) is constant for a small change in θ.

Therefore ρ always associates dφ , and r always associates dθ.

Differential Length (dl) :

- dl = dra_r + r dθa_θ + ρ dφ a_φ

We know that ρ = rsinθ

Therefore dl = dra_r + r dθa_θ + r sinθdφ a_φ

Differential length for a surface is given as:

- dl = dra_r + r dθa_θ ---( For r - θ plane or φ constant plane)

- dl = r dθa_θ + r sinθdφa_φ---( For θ - φ plane or r constant plane)

- dl = dra_r + r sinθdφ a_φ---( For r - φ plane or θ constant plane)

Differential length for a line parallel to r, θ and φ axis are respectively given as:

- dl = dra_r ---(For a line parallel to r axis)

- dl = r dθa_θ ---( For a line parallel to θ axis)

∫ dl = ∫_o^π rdθ = r (π - 0) = π r

This answer describes the circumference of a semi circle. Hence when θ varies with r and φ constant, the resultant length is the circumference of a semicircle.


- dl = r sinθdφa_φ ---( For a line parallel to φ axis)


Differential Surface (ds):

- ds = r dr dθa_φ

This surface describes a semi circle whose direction is along φ.

∫ ds = ∫_o^r ∫_o^π r drdθ 

= (r²/2) (π - 0) 

= π r²/2

This answer describes the surface area of a semicircle. Hence the surface represents a semicircle.


- ds = r² sinθdθ dφ a_r

This surface describes a hollow sphere or circumference or surface area of a sphere whose direction is along r.

∫ ds = ∫_o^2π ∫_o^π r²sin θ dθdφ 

= - r² (cosπ -cos 0) (2π -0) 

= 4 πr<sup>2

This answer describes the surface area of a sphere. Hence the surface is a sphere with its origin at the origin.</sup>


- ds = rsinθ drdφ a_θ

This describes a circular cone with z-axis as its axis and the origin at its vertex.

Differential Volume (dv):


dv = r² sinθ dr dθ dφ ---(Scalar Quantity)

∫ dv = ∫_o^r ∫_o^π ∫_o^2π r² sinθ drdθdφ 

= - (r³/3) (cosπ - cos 0^o) (2π - 0) 

= 4/3 (π r³)

This answer describes the volume of a sphere.

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) 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.

- 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.


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