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.

Cylindrical coordinate systems: In cylindrical coordinates, the surface are two planes and a cylinder. The surfaces used to define the cylindrical coordinates system are

(a) plane of constant z which is parallel to xy plane

(b) A cylinder of radius r with z axis as the axis of the cylinder

(c) A half plane perpendicular to xy plane and at an angle ϕ with respect to xz plane. The angle ϕ is called azimuthal angle.

Figure below shows the cylindrical coordinate system

A point P in space is represented by an order triple (r,ϕ,z). The angle ϕ is measured from the axis and expressed in radians. Anticlockwise measurement of ϕ is treated as positive while clockwise is treated as negative.

Conversion of cartesian to cylindrical coordinates.:

To convert the coordinate of a point from cartesian to cylindrical and viceversa let us consider point P having coordinates P(x,y,z) in cartesian and P(r,ϕ,z) in cylindrical system

Cartesian to cylindrical $(x,y,z) to(r,\Phi,z)$:

$r^2 = x^2 + y^2$

$tan ϕ =y/x$

z =z

Cylindrical to cartesian $(r,\Phi,z) to (x,y,z):$

$x = r cos ϕ$

$y = r sin ϕ$

z =z