In the figure an “infinite” sheet of positive charge density has been shown. (a) We have to calculate the work done by the electric field of the sheet as a small positive test charge is move from an initial position on the sheet to a final position located a perpendicular distance z from the sheet. (b) We have to use the result from (a) to show that the electric potential of an infinite sheet of charge can be written

where is the potential at the surface of the sheet.

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As we are considering an “infinite” nonconducting sheet electric field will be perpendicular to the plane of the sheet and its magnitude will be independent of the distance z from the sheet. By applying Gauss’ law, we have

or

As the charge on the sheet is positive, the direction of the electric field will be outward normal to the plane of the sheet.

(a)

Force on a test charge will be

Therefore, the work done by the field on moving charge from the surface of the sheet a distance z will be

.

(b)

Definition of electric potential is

.