22.12:
Electric Field of a Charged Disk
Consider a uniformly charged thin disk of radius R. The electric field at a point above its center at a perpendicular distance d is given by the surface integral over the disk.
Choose a ring of charge at a radius r, of width dr. Its area is the product of its circumference and width.
Let the electric field produced by any small charge element on the ring make an angle θ with the z-axis. Resolve this into two components: parallel and perpendicular to the z-axis.
There is a charge element whose electric field's perpendicular component is equal and opposite. So, the perpendicular components cancel. The disk's symmetry along its plane implies that it's the same for any such pair.
Since the z-components reinforce, the electric field points away from the disk. Integrating over r, the resultant field is obtained.
At large distances, the expression reduces to that of a point charge equal to the disk's total charge.
On the other hand, at small distances, the disk looks like an infinite plane whose electric field is constant.
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.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from 5.5 Calculating Electric Fields of Charge Distributions – University Physics Volume 2 | OpenStax; page 210.