Cylindrical Co-ordinate System

A co-ordinate system is used to define a point in space in a unique manner.

  • Cartesian system assumes that the space has length, width and breadth. So a point is defined by three distances from the reference point called “origin”. The three shortest distances when represented by straightlines are perpendicular to each other.
  • Cylindrical system considers:
    1. Shortest distance of the point from the vertical axis. This is known as “rho”.
    2. Shortest vertical distance of the point from the origin known as z.
    3. Angle made by the line joining the point and the vertical axis with reference x-axis. This is known as phi.
  • Spherical system considers:
  • Shortest distance of the point from the origin. This is known as “r”.
  • Angle by which the point is elavated from the horizontal plane. This is known as “theta”.
  • Angle by which the point is shifted from the x-axis. This is known as “phi”.

A simple model for cylindrical system

Cylindrical Co-ordinate System Model

The above image shows a simple model which helps to understand the cylindrical co-ordinate system. Anyone can make it using simple things.

Materials Used

  1. A cylindrical plastic bottle.
  2. A ballpen refill as shown in the image.
  3. A rubber-band.
  4. A toothpick.


You can assemble the cylindrical co-ordinate system model quickly. These are the steps:

  1. Drill small holes at the centres of lid and the bottom of the cylindrical bottle.
  2. Through these holes insert the ballpen refill as shown.
  3. Now attach the rubber band at the two ends of the refill as shown.
  4. On the bottle lid, mark reference axis -x with red colour and angle phi with black colour.
  5. Insert the toothpick where the rubber band turns to the cylindrical surface. The toothpick must be tangential to the cylinder lid as shown.
  6. Your model is ready.

To use the model

  1. Rotate the rubber band with respect to the refill (i.e. z-axis). This way you are changing the angle phi.
  2. Measure the angle between the rubber-band and the marked x-axis. This is phi.
  3. Distance of the point, where the toothpick is tangential (the blue spot), from the z-axis is “rho”. It is the length of the rubber band upto that point from z-axis.
  4. You can assume some point on z-axis to be the origin (here it is the green spot). Vertical distance of the point of contact of the toothpick from the origin is the z-coordinate.

This model is very useful to understand and visualise electrostatic and electromagnetic fields.

Please note

  • The direction of the “phi” axis is the direction of the tangent i.e. the toothpick.
  • The direction of the z-axis and x-axis are fixed.
  • At all points in space, these three axes are perpendicular to each other.
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