A FOUNTAIN PEN WORKS ON THE PRINCIPLE OF ATMOSPHERIC PRESSURE . IS THIS STATEMENT CORRECT ? PLEASE EXPLAIN .

Yes, it works on the principle of equilibrium between internal and external atmospheric pressures. Fountain pen has a reservoir in it to store ink into it. When one puts the nib to paper with light pressure, the split of the nib widens and brings ink stored in the pen through the feed to the split of the nib by capillary action and the flow of ink begins.
The nib of the pen take ink through the reservoir via gravity and capillary action. Capillary action under which with which the liquid to flow against the gravity and rise to the thin walls of the capillary. This involves the adhesion and cohesion forces. Due to this capillary action the flow of ink controls in the pen.

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