Physics 212 Lecture 8, Slide 1 Physics 212 Lecture 8 Today's Concept: Capacitors How does a capacitor behave in a circuit? More circuit examples

Main Point 1 First, filling the region between the plates of a capacitor with a dielectric will increase its capacitance. In particular, the ratio of the capacitance when filled with dielectric to that when filled with air is equal to kappa, the dielectric constant of the material. Physics 212 Lecture 8, Slide 2

Main Point 2 Second, the equivalent capacitance of two capacitors connected in parallel is equal to the sum of the individual capacitances. The voltage across each capacitor is the same and is the equivalent voltage of the combination. The equivalent charge of the combination is equal to the sum of the charges on the individual capacitors. Physics 212 Lecture 8, Slide 3

Main Point 3 Third, the equivalent inverse capacitance of two capacitors connected in series is equal to the sum of the individual inverse capacitances. The charge across each capacitor is the same and is the equivalent charge of the combination. The equivalent voltage of the combination is equal to the sum of the voltages across the individual capacitors. Physics 212 Lecture 8, Slide 4

Physics 212 Lecture 8, Slide 5 C V+Q-Q QQ This “Q” really means that the battery has moved charge Q from one plate to the other, so that one plate holds +Q and the other -Q. C V Q=VC Simple Capacitor Circuit 8

Physics 212 Lecture 8, Slide 6 C0C0C0C0 V Dielectrics C1=C0C1=C0C1=C0C1=C0 V

Physics 212 Lecture 8, Slide 7

Physics 212 Lecture 8, Slide 8 V C2C2C2C2 C1C1C1C1 Parallel Capacitor Circuit V C 12 C 1 + C 2 = C 12

Physics 212 Lecture 8, Slide 9 Series Capacitor Circuit C1C1C1C1 C2C2C2C2 V1V1V1V1 V2V2V2V2 Q V Q 1 C 12 1 C1C1C1C11 C2C2C2C2 + = V

Physics 212 Lecture 8, Slide 10 Checkpoint 1 C C C C C Which has lowest total capacitance : 18

Physics 212 Lecture 8, Slide 11 Checkpoint 2 Which has lowest total capacitance: C C C C C 20

Physics 212 Lecture 8, Slide 12 Checkpoint 3 25 A circuit consists of three unequal capacitors C 1, C 2, and C 3 which are connected to a battery of voltage V 0. The capacitance of C 2 is twice that of C 1. The capacitance of C 3 is three times that of C 1. The capacitors obtain charges Q 1, Q 2, and Q 3. Compare Q 1, Q 2, and Q 3. A. Q 1 > Q 3 > Q 2 B. Q 1 > Q 2 > Q 3 C. Q 1 > Q 2 = Q 3 D. Q 1 = Q 2 = Q 3 E. Q 1 < Q 2 = Q 3

Physics 212 Lecture 8, Slide 13 Checkpoint 4a/b 33 Two identical parallel plate capacitors are given the same charge Q, after which they are disconnected from the battery. After C 2 has been charged and disconnected, it is filled with a dielectric. Compare the voltages of the two capacitors. A. V 1 > V 2 B. V 1 = V 2 C. V 1 < V 2 Compare the potential energy stored by the two capacitors. A. U 1 > U 2 B. U 1 = U 2 C. U 1 < U 2

Physics 212 Lecture 8, Slide 14

Physics 212 Lecture 8, Slide 15 Checkpoint 4c Two identical parallel plate capacitors are given the same charge Q, after which they are disconnected from the battery. After C 2 has been charged and disconnected, it is filled with a dielectric. A. The charges will flow so that the charge on C 1 will become equal to the charge on C 2. B. The charges will flow so that the energy stored in C 1 will become equal to the energy stored in C 2. C. The charges will flow so that the potential difference across C 1 will become the same as the potential difference across C 2. D. No charges will flow. The charge on the capacitors will remain what it was before they were connected.

Physics 212 Lecture 8, Slide 16 In pre-lecture 7 we calculated the work done to move charge Q from one plate to another: Energy in a Capacitor (1/2 QV) +Q V -Q U = 1 / 2 QV C

Physics 212 Lecture 8, Slide 17

Physics 212 Lecture 8, Slide 18Calculation An air-gap capacitor, having capacitance C 0 and width x 0 is connected to a battery of voltage V. A dielectric (  ) of width x 0 /4 is inserted into the gap as shown. What is Q f, the final charge on the capacitor? Conceptual Analysis:  V C0C0 x0x0 V x 0 /4

Physics 212 Lecture 8, Slide 19 Calculation