Parallel-Plate Capacitor

Suppose that the distance d = r₂ - r₁ between the inner and outer sphere in Fig. 2-11 is held constant while the radii r₁ and r₂ are increased without limit. Then the ratio r₁/r₂ approaches unity.

When both sides of Eq. 2-28 are multiplied by the ratio r₁/r₂ the product is

[2-29]

As the ratio r₁/r₂ approaches unity the left-hand term of Eq. 2-29 approaches C₁₂ and Eq. 2-29 becomes reduced to

[2-30]

The quantity 4πr₁² in Eq. 2-30 represents the area of the dielectric, between the infinite spheres, normal to the electric field intensity. Hence, the capacitance per unit area of dielectric at right angles to the electric field, from Eq. 2-30, is

[2-31]

Since the voltage across the dielectric between spheres is V₁₂, the charge per unit area is

Then, for any area A between the infinite spheres and normal to the field, the charge is expressed by

The capacitance for this area A and thickness d of dielectric is found from Eq. 2-25 to be

[2-32]

Since the surfaces of the spheres approach those of planes as the radii r₁ and r₂ approach infinity, Eq. 2-32 expresses the capacitance between plane parallel plates of area A and separation d as shown in Fig. 2-13. Equation 2-32 is valid if there is no fringing of electric flux at the edges of the parallel plates. Fringing is appreciable unless the area A is large in relation to the separation d. Figure 2-14 shows a parallel-plate capacitor with flux fringing at the edges.

Figure 2-13. Parallel-plate capacitor

Figure 2-14. Flux fringing at the edges of parallel-plate capacitor