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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.

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.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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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 k 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 flled with a dielectric capacitor consists of two capacitors of thickness fd. connected in series. fd K (1 - Dd (1 - Dd C2 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 C2 fd C,- = Find the equivalent capacitance C for two capacitors combined in series: 1 1 1 fd Invert and substitute for the capacitance without the dielectric, Co A. d 1 fd K(1 - nd f+ x(1 - ) d KEO A KEO A C = f+ x(1 - ) d Finalize Let's test this result for some known limits. If f → 0, the dielectric should disappear. In this limit, c→ C, 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 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 x = 4, find the equivalent capacitance in pF.
Transcribed Image Text: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 k 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 flled with a dielectric capacitor consists of two capacitors of thickness fd. connected in series. fd K (1 - Dd (1 - Dd C2 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 C2 fd C,- = Find the equivalent capacitance C for two capacitors combined in series: 1 1 1 fd Invert and substitute for the capacitance without the dielectric, Co A. d 1 fd K(1 - nd f+ x(1 - ) d KEO A KEO A C = f+ x(1 - ) d Finalize Let's test this result for some known limits. If f → 0, the dielectric should disappear. In this limit, c→ C, 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 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 x = 4, find the equivalent capacitance in pF.
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