1. What is Spherical Coordinates Conversion?

Spherical coordinates (r, θ, φ) represent points in 3D space using radial distance and two angles. This system is particularly useful in physics and engineering for problems with spherical symmetry.

2. How Does the Calculator Work?

The calculator uses the following equations:

Where:

• \( r \) — Radial distance from origin
• \( \theta \) — Azimuthal angle in the xy-plane from x-axis
• \( \phi \) — Polar angle from z-axis
• \( x, y, z \) — Rectangular coordinates

Explanation: The equations transform Cartesian coordinates to a spherical coordinate system, which is often more natural for problems with spherical symmetry.

3. Importance of Coordinate Conversion

Details: Spherical coordinates simplify many physical problems, especially those involving central forces, wave propagation, or systems with spherical symmetry.

4. Using the Calculator

Tips: Enter x, y, z coordinates in meters. The calculator will output r (in meters), θ (azimuthal angle in degrees), and φ (polar angle in degrees).

5. Frequently Asked Questions (FAQ)

Q1: What's the range of θ and φ?
A: θ typically ranges from -180° to 180° (or 0° to 360°), while φ ranges from 0° to 180°.

Q2: How are singular points handled?
A: At the origin (0,0,0), r=0 and angles are undefined. Along the z-axis (x=y=0), θ is undefined.

Q3: What's the difference between θ and φ?
A: θ measures rotation around the z-axis (longitude), while φ measures angle from the z-axis (colatitude).

Q4: Why use spherical coordinates?
A: They simplify equations for systems with spherical symmetry, like atomic orbitals or gravitational fields.

Q5: How to convert back to rectangular coordinates?
A: Use x = r sinφ cosθ, y = r sinφ sinθ, z = r cosφ.