A Partially Filled Capacitor A parallel-plate capacitor with a plate separation d has a capacitance C, in the absence of a dielectric. What is the capacitance when a slab of dielectric material of dielectric constant x and thickness fd is inserted between the plates (see figure (a)), where f is a fraction between 0 and 1? (a) A parallel-plate capacitor of plate (b) The equivalent circuit of the separation d partially filled with a dielectric capacitor consists of two capacitors of thickness fd. connected in series. (1 - Dd (1 - nd C2 a SOLUTION Conceptualize In our previous discussions of dielectrics between the plates of a capacitor, the dielectric filled the volume between the plates. In this example, only part of the volume between the plates contains the dielectric material. Categorize In the example Effect of a Metallic Slab, we found that an infinitesimally thin metallic sheet inserted between the plates of a capacitor does not affect the capacitance. Imagine sliding an infinitesimally thin metallic slab along the bottom face of the dielectric shown in figure (a). We can model this system as a series combination of two capacitors as shown in figure (b). One capacitor has a plate separation fd and is filled with a dielectric; the other has a plate separation (Use the following as necessary: and has air between its plates. Analyze (Use the following as necessary: K, f, d, ao, and Co.) Evaluate the two capacitances in figure (b): and C, = fd Find the equivalent capacitance C for two capacitors combined in series: fd 1 + -- A. Invert and substitute for the capacitance without the dielectric, C. x(1 - nd f+ x(1 -0 d 1 fd + KE, A C = f+ x(1 -) d C = Finalize Let's test this result for some known limits. If f 0, the dielectric should disappear. In this limit, C Co which is consistent with a capacitor with air between the plates. If f 1, the dielectric fills the volume between the plates. In this limit, which of the following is correct? OC- Co Oc-K Oc- KC, EXERCISE A dielectric slab with thickness d is inserted halfway into the capacitor in the example (covering half the area). Given that d = 0.3 cm, A = 22 cm2, and k = 4, find the equivalent capacitance in pF.