Why Capacitor is used in a FAN

We all know what an electric fan is and know well that it works a capacitor but might be wondering how necessary it is to have capacitor along with a fan to get it work.
Honestly, it get me nervous if I have to look at the fan working but don't really know why capacitor must be used along with a fan to get it work so I decide to remove the capacitor from my ceiling fan and see if the fan will work but unfortunately for me, the fan didn't work instead just making a hum. I discover that I have to rotate it with my hand to get it rolling/working.
It sounds funny though but quite important for a typical electrical engineer student or workers to know the secret behind the usage of capacitor in the ceiling fan system.

What is a FAN?
An electric fan runs on AC power supply using capacitor as a means of starting device to kick-start the actual rotation of the fan. Now, lets observe the electrical operation of a fan. A fan is an asynchronous electric induction motor and from the study of electromagnetism and machine systems. Asynchronous motor are electric motor that require two source of electric supply in other to get started because of the stator and rotor inside the motor.
Stator the stationary part of the motor requires AC current to generate the require electromagnetic property
also, the Rotor which is the rotary part of this type of machine requires another source of electric current to generates its require electromagnetic property. This thus means that two source of electric current has to be supplied to each part of the motor component (stator and rotor) and the interaction of the two field created by this two device causes the require electromotive force needed to set the fan on motion.
This thus answer my first wondering as to why the fan rotates while the capacitor is removed and me using my hand to turn it which sets it into motion. 

So, If a capacitor is removed from a ceiling fan, it will not run, but it will rotate by pushing by hand because it get direction for magnetic repulsion

For this reason, a capacitor is used to provide another phase of current that actually leads the actual face of the supply incoming current of the fan.

So, in essence, ceiling fan is a single phase induction motor. These motors have low starting torque by themselves. So another phase is created, which is displaced from the main phase, and more this additional phase is displaced from the main phase, more is the starting torque of the motor. The capacitor helps to achieve exactly this. If you remove the capacitor, it would take forever for the fan to start rotating.

Summary

Ceiling fan uses a single phase induction motor to run it. This motor is NOT self-starting i.e. it requires external force to run it at the time of starting.
To overcome this drawback, a capacitor is used. It is connected across a part of the winding in the fan motor. This creates a phase difference between the windings (one part with the capacitor connected and other part without the capacitor).
Thus, a field is produced which is resultant of the two fields. The rotor rotates according to the direction of the resultant field.

The capacitor may be connected via. a centrifugal switch to increase the efficiency of the fan.