1 What is the spherical coordinate system? 

In this article I have explained what the spherical coordinate system is. Imagine you are trying to describe the location of a point in a room. You could say, “Move 2 metres along the floor, then 3 metres sideways, and 1 meter upward.”

That works well when the room has straight walls and everything is aligned with the x, y, and z directions. But now imagine you are standing around a satellite, antenna, electric charge, radar system, or spherical conductor. Suddenly, saying “move 2 metres in x, 3 metres in y…” doesn’t feel natural anymore.

Instead, an engineer may want to say, ‘How far am I from the source, what direction am I looking, and what angle am I at?’ That is exactly where the spherical coordinate system becomes useful.

2 Some Real-Life Applications of Spherical Coordinate Systems:

2.1 Field of Point Charge

Consider a simple point charge. A positive charge produces an electric field that spreads outward in every direction. An oscillating charge produces the radiations of a spherical pattern.

2.2 Spherical Coordinates in Antenna Engineering:

One of the best real-world applications is antenna engineering. An antenna does not simply radiate energy along the x-axis.

Electromagnetic energy spreads through three-dimensional space.

Engineers therefore need to know:

• How much power is radiated?
• In which direction is the radiation strongest?
• Where are the radiation nulls?
• What is the radiation pattern?
• How does the field vary with distance?
• What is the electric field at a particular angle?

For example, when designing a mobile-phone antenna, engineers analyse how electromagnetic waves propagate around the antenna.

The spherical coordinate system allows us to describe the field as a function of:

 

$$E(r,\theta,\phi)$$

 

So we can answer questions such as: “What is the electric field 20 meters away from the antenna at an angle of 45°?”

That’s much more intuitive using spherical coordinates.

2.3 Mobile phones and cellular towers

When you use your mobile phone, electromagnetic waves travel between your phone and a cellular base station.

The signal propagates through three-dimensional space.

Engineers need to understand:

• signal strength
• radiation pattern
• direction of propagation
• antenna gain
• electromagnetic field intensity
• coverage area

A cellular antenna may be mounted high above the ground and designed to radiate more strongly in some directions than others.

Spherical coordinates help engineers describe the 3D radiation pattern.

So the concept you learn in an electrical engineering mathematics course eventually appears in:

mobile communications → antenna design → wireless networks → 4G/5G systems.

2.4 Radar System and Spherical Coordinates

Radar is another excellent example. Imagine an airport radar system scanning the sky. It needs to determine where an aircraft is located. Instead of simply saying:

 

$$(x,y,z)$$

 

the radar naturally thinks in terms of:

How far away is the aircraft?

What is its elevation angle?

What is its azimuth angle?

That is essentially spherical-coordinate thinking.

The three quantities are closely related to:

 

$$r,\theta,\phi$$

 

This is why spherical coordinates are important in:

• radar engineering
• aerospace systems
• navigation
• defence electronics
• aircraft tracking
• satellite communication

3 How is a point represented in a spherical coordinate system?

In the Cartesian coordinate system, a point is represented by three components: (x,y,z), while a point P in space in spherical coordinates is uniquely represented in terms of (r,θ,ϕ). 

where:

• $r$ = distance from the origin
• $\theta$ = polar angle measured from the $+z$-axis
• $\phi$ = azimuthal angle measured around the $z$-axis

So instead of describing a point using three straight-line distances, we describe it using:

distance + direction + angle

This becomes extremely powerful when the electrical phenomenon we are studying spreads outward from a point or has a spherical shape.

The surface r = constant is a sphere,

The surface θ = constant is a cone,

(two surfaces, cone and sphere, are everywhere perpendicular along their intersection, which is a circle of radius r sin θ)

The surface φ = constant is a plane passing through the θ = 0 line (or the z axis). The projection of r on x-plane is OM=r sin⁡θ.”The positive direction of ” θ” is from the positive z-axis, where its value is 0, towards the negative z-axis, where its value is π. So ” θ” varies from 0 to π. “However, 0≤r≤∞.

These three surfaces are mutually perpendicular to each other, as can be seen in the figure below.

We consider any point as the intersection of three mutually perpendicular surfaces—a sphere, a cone, and a plane—each orientated in the manner just described.

Unit Vectors in Spherical Coordinate System

The unit vector ar is directed radially outward, normal to the sphere r = constant, and lies in the cone θ = constant and the plane φ = constant.

The unit vector aθ is normal to the conical surface, lies in the plane, and is tangent to the sphere.

The third unit vector aφ is the same as in cylindrical coordinates, being normal to the plane and tangent to both the cone and the sphere.

Relation Between Spherical and Rectangular Coordinate Systems

The relation between spherical and rectangular coordinate systems can be found using the following formulas.

\[
\boxed{
\begin{aligned}
x &= r\sin\theta\cos\phi \\[4pt]
y &= r\sin\theta\sin\phi \\[4pt]
z &= r\cos\theta
\end{aligned}}
\]

\[
\boxed{
r = \sqrt{x^2+y^2+z^2}
}
\]

\[
\boxed{
\theta = \cos^{-1}\left(\frac{z}{r}\right)
}
\]

\[
\boxed{
\phi = \tan^{-1}\left(\frac{y}{x}\right)
}
\]

Differential Element in Spherical Coordinate System

A differential volume element may be constructed in spherical coordinates by increasing r , θ, and φ by dr, dθ, and dφ, as shown in figure below:

Area of Differential Surfaces

In spherical coordinates, a point is represented by $(r,\theta,\phi)$, where $r$ is the radial distance, $\theta$ is the polar angle, and $\phi$ is the azimuthal angle. A small differential element has dimensions $dr$, $r\,d\theta$, and $r\sin\theta\,d\phi$.

Thus, the differential volume element is:

\[
dV = r^2\sin\theta \, dr\,d\theta\,d\phi
\]

Differential Surface Areas

For a surface of constant $r$:

\[
dS_r = r^2\sin\theta \, d\theta\,d\phi
\]

\[
dS_r = r^2\sin\theta \, d\theta\,d\phi
\]

For a surface of constant $\theta$:

\[
dS_\theta = r\sin\theta \, dr\,d\phi
\]

For a surface of constant $\phi$:

\[
dS_\phi = r\,dr\,d\theta
\]

Total Volume

For a spherical region,

\[
V = \int_0^{2\pi}\int_0^\pi\int_0^R
r^2\sin\theta \, dr\,d\theta\,d\phi
\]

\[
V = \frac{4}{3}\pi R^3
\]

Surface Area of a Sphere

For a sphere of radius $R$:

\[
A = \int_0^{2\pi}\int_0^\pi
R^2\sin\theta \, d\theta\,d\phi
\]

\[
A = 4\pi R^2
\]

A Complete Summary of Differential Elements Surface Areas and Volume in Spherical Coordinate

Electromagnetic Field Theory lecture 3

Video Lecture of Spherical Coordinate System

The follow lecture describes the complete detail of spherical coordinate systems.

Relation between spherical and cartesian coordinates systems

https://www.youtube.com/watch?v=zcoOkG7ppV0

Also read here

https://eevibes.com/electromagnetic-field-theory/what-is-the-cylindrical-coordinate-system/

what is the cylindrical coordinate system?