Subject :- Applied Physics 2.

Topic :- Laser.

Difficulty :- Medium.

In 3 Dimension, the coordinate system can be specified by the intersection of three surfaces. Three commonly used right handed orthogonal coordinate systems are-

1. Cartesian or rectangular coordinate system.

2.Cylindrical or circular coordinate system.

1. spherical or polar coordinate system.

Cartesian coordinate system: In the cartesian coordinate system all of the three surfaces are planes. The three coordinate axes x,y,and z are mutually perpendicular to each other. In this system, a point P in space is represented by an order triple(x,y,z). These are respectively the distances from the origin to the intersection of a perpendicular dropped from the point P to x,y,z axes as shown in the diagram.the coordinates x,y,and z can be positive or negative.

Spherical coordinate system: In spherical coordinates, the surface are a sphere , a plane and a cone. In this coordinate system a point P in space is represented by an order triple (r,ϴ,ϕ). The first coordinate is a distance, and the second and third coordinates are angles.

The surface which are used to define the spherical coordinate system on three cartesian axes are

(a) Sphere of radius r (or radial distance), origin as the center of the sphere.

(b) A right circular cone with its apex at the origin and its axes as z axis.Its half angle is ϴ (known as polar angle).It rotates about z axis and ϴ varies from 0° to 180°.

(c) A half plane perpendicular to xy plane containing z axis, making an angle ϕ (known as azimuthal angle) with xz plane.

Transformation of cartesian to spherical: (x,y,z) to (r,ϴ,ϕ )

$$ r=\sqrt{x^2+y^2+z^2}\\ \Theta=\tan^{-1}\frac{\sqrt{x^2+y^2}}{z}\\ \Phi=\tan^{-1}\frac{y}{x}$$

Transformation of spherical to cartesian: (r,ϴ,ϕ ) to (x,y,z)