23.5 The Parallel-Plate Capacitor

FIGURE 23.19 shows two metal plates called electrodes, one with charge +Q and the other with −Q, placed face-to-face a distance d apart. This arrangement of two electrodes, charged equally but oppositely, is called a parallel-plate capacitor. Capacitors play important roles in many electric circuits. Our goal is to find the electric field both inside the capacitor (i.e., between the plates) and outside the capacitor.

FIGURE 23.19 A parallel-plate capacitor.

The figure illustrates a parallel-plate capacitor.
FIGURE 23.19 Full Alternative Text

NOTE   The net charge of a capacitor is zero. Capacitors are charged by transferring electrons from one plate to the other. The plate that gains N electrons has charge −Q=N(−e). The plate that loses electrons has charge +Q. The symbol Q that we will use in equations refers to the magnitude of the charge on each plate.

FIGURE 23.20 is an enlarged view of the capacitor plates, seen from the side. We will assume that the plate separation d is much smaller than their diameter, so the “broken off” look at the top and bottom indicates that the plates extend far above and below what we see in the figure. Because opposite charges attract, all of the charge is on the inner surfaces of the two plates. Thus the inner surfaces of the plates can be modeled as two planes of charge with equal but opposite surface charge densities. As you can see from the figure, at all points in space the electric field E⃗+ of the positive plate points away from the plane of positive charges. Similarly, the field E⃗− of the negative plate everywhere points toward the plane of negative charges.

FIGURE 23.20 The electric fields inside and outside a parallel-plate capacitor.

The figure illustrates electric fields inside and outside a parallel-plate capacitor.
FIGURE 23.20 Full Alternative Text

NOTE   You might think the right capacitor plate would somehow “block” the electric field created by the positive plate and prevent the presence of an E⃗+ field to the right of the capacitor. To see that it doesn’t, consider an analogous situation with gravity. The strength of gravity above a table is the same as its strength below it. Just as the table doesn’t block the earth’s gravitational field, intervening matter or charges do not alter or block an object’s electric field.

Outside the capacitor, E⃗+ and E⃗− point in opposite directions. Because the field of a plane of charge is independent of the distance from the plane, the fields also have equal magnitudes. Consequently, the fields E⃗+ and E⃗− add to zero outside the capacitor plates. There’s no electric field outside an ideal parallel-plate capacitor.

Inside the capacitor, between the electrodes, field E⃗+ points from positive to negative and has magnitude η/2ϵ0=Q/2ϵ0A, where A is the surface area of each electrode. Field E⃗− also points from positive to negative and also has magnitude Q/2ϵ0A, so the inside field E⃗++E⃗− is twice that of a plane of charge. Thus

E⃗capacitor={ (Qϵ0A, from positive to negative)​inside0→outside (23.31)

FIGURE 23.21a shows the electric field—this time using field lines—of an ideal parallel-plate capacitor. No real capacitor is infinite in extent, but the ideal parallel-plate capacitor is a very good approximation for all but the most precise calculations as long as the electrode separation d is much smaller than the electrodes’ size. FIGURE 23.21b shows that the interior field of a real capacitor is virtually identical to that of an ideal capacitor but that the exterior field isn’t quite zero. This weak field outside the capacitor is called the fringe field. We will always assume that the plates are very close together and use Equation 23.31 for the field inside a parallel-plate capacitor.

FIGURE 23.21 Ideal versus real capacitors.

The figure labeled (“a”) illustrates the edge view of an ideal capacitor.
The figure labeled (b) illustrates the edge view of a real capacitor.
Full Alternative Text

NOTE   The shape of the electrodes—circular or square or any other shape—is not relevant as long as the electrodes are very close together.

EXAMPLE 23.6 The electric field between the plates of a capacitor

Two 1.0 cm×2.0 cm rectangular electrodes are 1.0 mm apart. What charge must be placed on each electrode to create a uniform electric field of strength 2.0×106 N/C? How many electrons must be moved from one electrode to the other to accomplish this?

MODEL The electrodes can be modeled as an ideal parallel-plate capacitor because the spacing between them is much smaller than their lateral dimensions.

SOLVE The electric field strength between the plates of a capacitor is E=Q/ϵ0A. Thus the charge to produce a field of strength E is

Q=(8.85×10−12 C2/N m2)(2.0×10−4 m2)(2.0×106 N/C)=3.5×10−9 C=3.5 nC

The positive plate must be charged to +3.5 nC and the negative plate to −3.5 nC. In practice, the plates are charged by using a battery to move electrons from one plate to the other. The number of electrons in 3.5 nC is

N=Qe=3.5×10−9 C1.60×10−19 C/electron=2.2×1010 electrons

Thus 2.2×1010 electrons are moved from one electrode to the other. Note that the capacitor as a whole has no net charge.

REVIEW The plate spacing does not enter the result. As long as the spacing is much smaller than the plate dimensions, as is true in this example, the field is independent of the spacing.