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ParallelPlate Capacitor
Suppose that the distance d = r_{2}  r_{1} between the inner and outer sphere in Fig. 211 is held constant while the radii r_{1} and r_{2} are increased without limit. Then the ratio r_{1}/r_{2} approaches unity. When both sides of Eq. 228 are multiplied by the ratio r_{1}/r_{2} the product is
As the ratio r_{1}/r_{2} approaches unity the lefthand term of Eq. 229 approaches C_{12} and Eq. 229 becomes reduced to
The quantity 4πr_{1}^{2} in Eq. 230 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. 230, is
Since the voltage across the dielectric between spheres is V_{12}, 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. 225 to be
Since the surfaces of the spheres approach those of planes as the radii r_{1} and r_{2} approach infinity, Eq. 232 expresses the capacitance between plane parallel plates of area A and separation d as shown in Fig. 213. Equation 232 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 214 shows a parallelplate capacitor with flux fringing at the edges.


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