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First of all, capacitor is added to the additional auxiliary winding of stater. As a result, it creates a field which leads by 90 degree to main winding field. Therefore, we get two fields which are 90 degrees to each other. Building on this, a rotating field responsible for start of the fan motor. So, that is why we us a capacitor in a ceiling fan.
In addition to this, the fan has an AC single phase motor. The problem with single phase induction motor is that there is only one phase on the stator winding. Hence, the resulting magnetic field in a single phase motor does not rotate. For starting any type of motor there must be resulting magnetic field to create torque to rotate the rotor .In single phase motor without capacitor the field created by stator winding is positive in half cycle and negative in other half so resulting net field is zero so torque is also zero. Now, when a capacitor is added to the additional auxiliary winding of stator it creates field which leads by 90 degree to main winding field so we get two fields which are 90 degrees to each other, resulting a rotating field responsible for start of the fan motor. The capacitor used to start the motor after start may be taken out of circuit because as rotor is rotating each half field of stator creating field in rotor in same direction, so producing net field not equal to zero and motor can run continuously if we do not switch off the power supply to the motor
