Many electronic devices use charged, flat surfaces—disks, squares, rectangles, and so on—to steer electrons along the proper paths. These charged surfaces are called electrodes. Although any real electrode is finite in extent, we can often model an electrode as an infinite plane of charge. As long as the distance z to the electrode is small in comparison to the distance to the edges, we can reasonably treat the edges as if they are infinitely far away.

The electric field of a plane of charge is found from the on-axis field of a charged disk by letting the radius $R\to \infty$. That is, a disk with infinite radius is an infinite plane. From Equation 23.24, we see that the electric field of a plane of charge with surface charge density $\eta$ is

This is a simple result, but what does it tell us? First, the field strength is directly proportional to the charge density $\eta :$ more charge, bigger field. Second, and more interesting, the field strength is the same at all points in space, independent of the distance z. The field strength 1000 m from the plane is the same as the field strength 1 mm from the plane.

How can this be? It seems that the field should get weaker as you move away from the plane of charge. But remember that we are dealing with an infinite plane of charge. What does it mean to be “close to” or “far from” an infinite object? For a disk of finite radius R, whether a point at distance z is “close to” or “far from” the disk is a comparison of z to R. If $z\ll R$, the point is close to the disk. If $z\gg R$, the point is far from the disk. But as $R\to \infty$, we have no scale for distinguishing near and far. In essence, every point in space is “close to” a disk of infinite radius.

No real plane is infinite in extent, but we can interpret Equation 23.28 as saying that the field of a surface of charge, regardless of its shape, is a constant $\eta \text{/}2{ϵ}_0$ for those points whose distance z to the surface is much smaller than their distance to the edge.

We do need to note that the derivation leading to Equation 23.28 considered only $z>0$. For a positively charged plane, with $\eta >0$, the electric field points away from the plane on both sides of the plane. This requires $E_z<0$ ($\vec{E}$ pointing in the negative z-direction) on the side with $z<0$. Thus a complete description of the electric field, valid for both sides of the plane and for either sign of $\eta$, is

The infinite plane of charge is the third of our important electric field models.

FIGURE 23.17 shows two views of the electric field of a positively charged plane. All the arrows would be reversed for a negatively charged plane. It would have been very difficult to anticipate this result from Coulomb’s law or from the electric field of a single point charge, but step by step we have been able to use the concept of the electric field to look at increasingly complex distributions of charge.

FIGURE 23.17 Two views of the electric field of a plane of charge.

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