22.12: Electric Field of a Charged Disk
The simplest case of a surface charge distribution is the uniformly charged disk. Calculating its electric field also helps us calculate the electric field of a large plane of charge.
The system's symmetry is in the cylindrical directions across the plane of the charge. As a result, the electric fields created by various surface charge elements nullify each other in the direction parallel to the surface. Thereby, the resulting electric field is perpendicular to the plane. Since the disk is symmetrical on its two faces, the electric field is perpendicular to the plane on both sides. As a result, if the charge density is positive, the field points away from the plane on both sides and if the charge density is negative, it points into the plane on both sides.
Symmetry is best understood by considering a ring of charge at any radius. Similar pairs exist across it, which produce equal but opposite electric fields at a point above the disk's origin.
Since the principle of superposition of electric fields holds for each component, the fields along the perpendicular direction reinforce each other, simplifying its calculation.
The charged disk is expected to behave like a point charge, of total charge equal to the total charge of the disk, at a large distance, where the internal charge distribution is irrelevant. That is indeed what happens.
It is interesting to note what happens at small distances when the field point being probed is very close to the disk. Here, the disk looks like a plane stretching to infinity. The small-distance approximation of the electric field implies that the electric field is constant. That is, very close to the disk, the field does not depend on the distance. This approximation is invoked in the case of parallel plate capacitors.
