In electrical engineering and physics, the Spherical Coordinate System is used to locate a point in 3D space using one distance and two angles.
A point in this system is defined as \((r, \theta, \phi)\).
The three coordinates are as follows:
- \(r\) (Radial Distance): The direct distance from the origin \((O)\) to the point \((P)\).
Range: \(0 \leq r < \infty\) - \(\theta\) (Colatitude / Polar Angle): The angle measured down from the positive z-axis.
Range: \(0 \leq \theta \leq \pi\) (or \(0^\circ\) to \(180^\circ\)) - \(\phi\) (Azimuthal Angle): The angle measured from the positive x-axis in the x-y plane.
Range: \(0 \leq \phi < 2\pi\) (or \(0^\circ\) to \(360^\circ\))
Conversion from Spherical to Cartesian coordinates:
If \((r, \theta, \phi)\) are known, then the Cartesian coordinates \((x, y, z)\) can be obtained using:
- \(x = r \sin\theta \cos\phi\)
- \(y = r \sin\theta \sin\phi\)
- \(z = r \cos\theta\)
Conversely, the radial distance from Cartesian coordinates is: \(r = \sqrt{x^2 + y^2 + z^2}\).
Differential elements (important in calculus):
- Differential length:
\(dl = dr \hat{a}_r + r d\theta \hat{a}_\theta + r \sin\theta d\phi \hat{a}_\phi\) - Differential surface area (for a sphere where \(r\) is constant):
\(dS = r^2 \sin\theta \, d\theta \, d\phi\) - Differential volume:
\(dV = r^2 \sin\theta \, dr \, d\theta \, d\phi\)
Quick Summary Table:
| Component | Name | Measured From | Range |
|---|---|---|---|
| \(r\) | Radius | Origin | \(0\) to \(\infty\) |
| \(\theta\) | Polar Angle | \(+z\) axis | \(0^\circ\) to \(180^\circ\) |
| \(\phi\) | Azimuthal Angle | \(+x\) axis | \(0^\circ\) to \(360^\circ\) |
Why it is used in Electrical Engineering:
- Electrostatics: The electric field of a point charge is given by
\(E = \frac{Q}{4\pi\epsilon r^2} \hat{a}_r\), which depends only on \(r\), making spherical coordinates very convenient. - Antennas: Radiation patterns are typically expressed in terms of
\(\theta\) and \(\phi\), especially in far-field analysis.