The Parallel Plate Capacitor

Parallel plate capacitor

The figure above depicts a parallel plate capacitor. Here, we can see two large plates placed parallel to each other at a small distance d. The distance between the plates is filled with a dielectric medium as shown by the dotted array. The two plates carry an equal and opposite charge. Here, we see that the first plate carries a charge +Q and the second carries a charge –Q. The area of each of the plates is A and the distance between these two plates is d. The distance d is much smaller than the area of the plates and we can write d<<A, thus the effect of the plates are considered as infinite plane sheets and the electric field generated by them is treated as that equal to the electric field generated by an infinite plane sheet of uniform surface charge density. As the total charge on plate 1 is Q and the area of the plate is A, the surface charge density can be given as

Similarly, for the plate 2 with total charge equal to –Q and area A, the surface charge density can be given as,

We divide the regions around the parallel plate capacitor into three parts, with area 1 being the area left to the first plate, area 2 being the area between the two planes and area 3 is the area to the right of plate 2.

Let us calculate the electric field in the region around a parallel plate capacitor.

Region I: The magnitude of electric field due to both the infinite plane sheets I and II is the same at any point in this region, but the direction is opposite to each other, the two forces cancel each other and the overall electric field can be given as,

Region II: The magnitude, as well as the direction of electric field due to both the plane sheets I and II in this regions, is the same and the overall effect can be given as,

Region III:  Similar to region I, here too the magnitude of the electric field generated due to both the plane sheets I and II is the same but the direction is opposite, giving the same result as,

Here, the electric field is uniform throughout and its direction is from the positive plate to the negative plate.

The potential difference across the capacitor can be calculated by multiplying the electric field and the distance between the planes, given as,

And the capacitance for the parallel plate capacitor capacitor can be given as,