Radius of the plates of capacitor = R
Distance between the capacitor = d<<R
Radius of conducting disk = r<<R
Thickness of conducting disk = t<<r
When the conducting disk is in contact with the bottom plate of the capacitor, then it will get charged from the capacitor plate, due to which a force will be exerted on the disk due to the electric field present between the plates of the capacitor. Therefore, when this force due to the electric field equals the gravitational force on the disk due its own mass m, the disk will be lifted.
We know that the electric field and potential are related by the equation,
To find the charge pass on to the conducting disk we need to find out the charge flowing between the plates due to the electric field through an area equal to the area of the conducting disk. For this we apply the Gauss’s law. Gauss’s law states that the net electric flux enclosed through a closed surface is equal to the charge enclosed within the surface divided by the permittivity i.e.
Where q is the charge enclosed within the surface.
Now assuming an imaginary cylinder of radius r with the conducting disk inside it, perpendicular to the axis as shown in the figure, the charge transferred to the disk can be calculated as follows:
Since the electric field is perpendicular to the disk and parallel to the surface of the cylinder, therefore the angle between the surface normal and electric field is , at the ends of the cylinder and at the curved surface, therefore
Now force on the conducting disk due to the electric field
Now for minimum voltage this force should at least be equal to the force due to gravity i.e.
For minimum V,
Which is the minimum voltage required to life the disk from the plate.
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