The Spherical Coordinate System - Vector Analysis - Electromagnetic Field and Wave Theory | Summary and Q&A

TL;DR

Spherical coordinate system is used to represent points in a three-dimensional space using a radius and two angles, providing a different perspective from rectangular and cylindrical coordinate systems.

Key Insights

• 😥 Spherical coordinates provide a different perspective for representing points in three-dimensional space, particularly on curved surfaces.
• 😥 Spherical coordinate system has a single reference point, the origin, for representing points.
• 👻 The range of values for θ and φ in spherical coordinates allows for the representation of various shapes and planes.
• ❓ Converting between rectangular and spherical coordinates involves using trigonometric functions and algebraic calculations.
• 👲 Unit vectors, such as a r cap, a theta cap, and a phi cap, are used to denote the directions of increasing values for r, θ, and φ, respectively.
• 🫥 Transforming vectors between different coordinate systems often involves using dot product operations and coordinate system-specific unit vectors.

Transcript

click the bell icon to get latest videos from ekeeda hello everyone in the previous lectures we have gone through rectangular coordinate system and cylindrical coordinate system while we switch from rectangular coordinate system to cylindrical coordinate system when angle was introduced that was fine in the case of spherical coordinate system now t... Read More

Questions & Answers

Q: What are the three main components of a point represented in the spherical coordinate system?

The three main components are the radius (r), the angle θ, and the angle φ. The radius represents the distance from the origin to the point, while θ represents the angle made by the radius with the positive z-axis, and φ represents the angle made by the radius with the positive x-axis.

Q: How are planes associated with the spherical coordinate system?

Spherical coordinate system allows for the representation of different planes. A constant value of r represents a spherical surface, a constant value of θ represents a conical surface, and a constant value of φ represents a vertical plane in a specific orientation.

Q: What is the range of values for the angles θ and φ in the spherical coordinate system?

The angle θ ranges from 0 to 180 degrees, as it can extend upwards, downwards, left, right, front, and back. The angle φ ranges from 0 to 360 degrees, covering a full circle in the x-y plane.

Q: How can points be converted from rectangular coordinates to spherical coordinates?

To convert from rectangular coordinates (x, y, z) to spherical coordinates (r, θ, φ), we can use the following formulas:

• r = √(x^2 + y^2 + z^2)

• θ = cos^(-1)(z / √(x^2 + y^2 + z^2))

• φ = tan^(-1)(y / x) + 180° if x < 0

Summary & Key Takeaways

• Spherical coordinate system uses a radius (r) and two angles (θ and φ) to represent points in three-dimensional space.

• θ represents the angle between the radius vector and the positive direction of the z-axis, while φ represents the angle between the radius vector and the positive direction of the x-axis.

• Spherical coordinate system allows for the representation of points on spherical surfaces, conical surfaces, and vertical planes.