Chapter 21 Electric Current and Direct-Current Circuit

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1 Chapter 2 Electric Current and Direct-Current Circuit Outline 2- Electric Current 2-2 Resistance and Ohm s Law 2-3 Energy and Power in Electric Circuit 2-4 Resistance in Series and Parallel 2-5 Kirchhoff s Rules 2-6 Circuits containing Capacitors 2-7 RC Circuits

2 2-6 Circuits containing Capacitors Capacitors in Parallel Capacitor are connected in parallel, and they have the same voltage. Figure 2-6 shows three capacitors connected in parallel. Figure 2-6 Capacitors in Parallel

3 Deriving equivalent capacitance In fig. 2-6 (a), the magnitudes of the charges on each capacitor are Q C, Q C, Q C, The total charge on the three capacitors is Q Q Q Q C C C ( C C C ) () In fig. 2-6 (b), the magnitude of the charge on an equivalent capacitor is Q C eq (2) Compared the above two equations () and (2), we have C C C C eq 2 3

4 Equivalent capacitance for Capacitors in Parallel Ceq C C2 C3... C (2 4) SI unit: farad, F

5 Problem 2-56 capacitor in parallel Two capacitors, one 7.5 µf, and the 5µF, are connected in parallel across a 2-V battery. (a) Find the equivalent capacitance of the two capacitors. (b) Find the charge stored in each capacitor.

6 Solution ) Since the 2 capacitors in parallel: C eq C C eq C F 22.5 F 5 F 2) Both capacitors have the same voltage 2 V: Q Q C 2( V ) ( F) C Q 2 Q C 2 2( V ) (5 0 6 F) C

7 Capacitors in Series Capacitor are connected in series (one after the other), and they have the same charge. Figure 2-7 shows three capacitors connected in series. Figure 2-7 Capacitors in Series

8 Deriving equivalent capacitance In figure (a), since all the capacitors have the same charge Q, we have Q Q Q V, V, V, 2 3 C C2 C3 And the total potential difference across the three capacitors must equal the battery emf, Q Q Q V V2 V3 Q( ) (2 5) C C C C C C In figure (a), since Q = C eq ε, we have Q Ceq Compare (2-5) and (2-6), one has C C C C eq 2 3 (2 6)

9 Equivalent capacitance for Capacitors in Series... (2 7) C C C C C eq 2 3 SI unit: farad, F

10 Problem 2-56 capacitor in series Two capacitors, one 7.5 µf, and the 5µF, are connected in series across a 2-V battery. (a) Find the equivalent capacitance of the two capacitors. (b) Find the charge stored in each capacitor.

11 Solution ) Since the 2 capacitors in series: C eq C C C eq eq C F 5 F F 0.20 ( F) 2) Both capacitors have the same voltage 2 V: Q Q 2 Q 2( V ) (5 0 6 F) C

12 Active Example 2-3 Find Equivalent Capacitance A electric circuit consists of a 2.0 V battery and three capacitors connected partly in series and partly in parallel. Find the total energy store in the capacitors. Active Example 2-3 Find the Equivalent Capacitance and the Stored Energy

13 Solution ) Find the equivalent capacitance of a 0.0 uf and 5.00 uf in series, 3.33 uf 2) Find the equivalent capacitance of circuit (3.33 uf and 20.0 uf in parallel) C eq = 23.3 uf 3) Find the stored energy U C V J 2 eq

14 2-7 RC Circuit A RC circuit consists of a resistor R and capacitor C. Figure 2-8 A Typical RC Circuit Charging the circuit!!

15 The formula that can be used to calculate the charge on the capacitor with time t as a parameter is, qt C e t / () ( ) (2 8), where, e is exponential. = RC is called Time Constant (s: second), which determines the property of the CR circuit. When t =, q = Cε (63.2% of the final charge). Figure 2-9 Charge Versus Time for an RC Circuit

16 The current as a function of time t can be expressed as It R t / () ( ) e (2 9), = RC is called Time Constant (s: second), Figure 2-20 Current Versus Time in an RC Circuit

17 Example 2-9 Charging a Capacitor A circuit consists of a 26-Ω resistor, a 275-Ω resistor, a 82-uF capacitor, a switch, and a 3.00-V battery all in series. Initially the capacitor is uncharged and the switch is open. At time t = 0, the switch is closed. (a) What charge will the capacitor have a long time after the switch is closed? (b) At what time will the charge on the capacitor be 80.0% of the value found in part (a)? Picture the problem Example 2-9 Charging a Capacitor

18 Solution Part (a) Time constant ( )(82 0 ) ) After a long time ( t ), t / qt () C ( e ) C 6 3 RC F s 6 6 (82 0 F)(3.00 V) C 2) Set q(t) = Cε, and solving for t, We have t / qt ( ) C ( e ) 0.8C t / e 0.8 t ln(0.200) so, 3 3 t ln(0.200) ( s) ln(0.200) 8 0 s

19

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21 Summary Equivalent capacitance for Capacitors in Parallel Ceq C C2 C3... C (2 4) Equivalent capacitance for Capacitors in Series... (2 7) C C C C C eq RC Circuit 2 3 qt C e t / () ( ) (2 8),

22 Example 2-8 Energy in Parallel Two capacitors, one 2.0 uf, and the other of unknown capacitance C, are connected in parallel across a battery with an emf 9.00 V. The total energy stored in the two capacitors is 0.05J. What is the value of the capacitance C. Example 2-8 Energy in Parallel

23 Solution ) Since the Capacitor energy is 2 U QV C V 2 2 eq We have J Ceq (9.00 V) 2 6 C F 284 F eq 2) Solve for C in terms of equivalent capacitance: C C C eq 284 F 2.0 F C C 284 F 2.0 F 272 F

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