Spherical Coordinate System (r, θ, φ) - Field Theory

Geographers specify a location on the Earth’s surface using three scalar values:
longitude, latitude, and altitude.

Both longitude and latitude are angular measures, while altitude is a measure of distance.

Latitude, longitude, and altitude are similar to Spherical Co-ordinates.

Spherical coordinates consist of one scalar value (r), with units of distance, while the other two scalar values (θ, φ) have angular units (degrees or radians).

A Vector in Spherical System is represented as
(A_r A_θ, A_φ)
or
A = A_ra_r + A_θa_θ + A_φa_φ

Where a_r, a_θ and a_φ are the unit vectors in r, θ and φ direction respectively.

The physical significance of each parameter of spherical coordinates:


- The value r expresses the distance of the point from origin (i.e. similar to altitude). It is the radius of the sphere.

- The angle θ is the angle formed with the z- axis (i.e. similar to latitude). It is also called the co-latitude angle. It is measured clockwise.

- The angle φ, also called the azimuthal angle, indicates the rotation angle around the z-axis (i.e. similar to longitude). It is basically measured from the x axis in the x-y plane. It is measured counter-clockwise.


Range of the variables:

It defines the minimum and the maximum value that r, θ and φ can have in spherical co-ordinate system.

- 0 ≤ r ≤ ∞
- 0 ≤ θ ≤ π
- 0 ≤ φ ≤ 2π

Spherical System - Unit Vectors:

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

- a_r points in the direction of increasing r i.e.
    a_r points away from the z-axis.
- _aθ points in the direction of increasing θ.
- a_φ points in the direction of increasing φ.

Consider the parallelogram ABOC,

- X = ρcos φ.
- Y = ρsin φ.
- Z = Z.

Consider the second parallelogram OCPQ, we have

- ρ = r sin θ
- z = r cos θ

So from the above data available we can say,

- x = r sin θ cos φ.
- Y = r sin θ sin φ.
- Z = r cos θ.

Similarly relationship between spherical and cylindrical coordinates can be derived as:

Relationship between (a_x, a_y, a_z) and (a_r, a_θ, a_φ)

From the cylindrical coordinate system we know that,

a_x = cosφ a_ρ – sinφ a_φ
a_y = sinφ a_ρ + cosφ a_φ
a_z = a_z

From the above figure, we can write a_ρ in terms of a_r and a_θ as

a_ρ = cos (90^o – θ) a_r + cos θ a_θ
a_z = cos θ a_r - sin θ a_θ  

Hence the unit vectors of cartesian and spherical co-ordinate system are related as:


a_x = sin θ cos φ a_r + cos θ cos φ a_θ - sin φ a_φ
a_y = sin θ sin φ a_r + cos θ sin φ a_θ + cos φ a_φ
a_z = cos θ a_r - sin θ a_θ



Transformation of vector A from (A_r, A_θ, A_Ф) to (A_x, A_y, A_z) i.e. transformation of Vector A from Spherical to Cartesian can be obtained as




Transformation of vector A from (A_x, A_y, A_z) to (A_r, A_θ, A_Ф) i.e. transformation of Vector A from Cartesian to Spherical can be obtained as






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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