Chapter 21 Electric Current and DirectCurrent Circuit


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1 Chapter 2 Electric Current and DirectCurrent Circuit Outline 2 Electric Current 22 Resistance and Ohm s Law 23 Energy and Power in Electric Circuit 24 Resistance in Series and Parallel 25 Kirchhoff s Rules 26 Circuits containing Capacitors 27 RC Circuits
2 26 Circuits containing Capacitors Capacitors in Parallel Capacitor are connected in parallel, and they have the same voltage. Figure 26 shows three capacitors connected in parallel. Figure 26 Capacitors in Parallel
3 Deriving equivalent capacitance In fig. 26 (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. 26 (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 256 capacitor in parallel Two capacitors, one 7.5 µf, and the 5µF, are connected in parallel across a 2V 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 27 shows three capacitors connected in series. Figure 27 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 (25) and (26), 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 256 capacitor in series Two capacitors, one 7.5 µf, and the 5µF, are connected in series across a 2V 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 23 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 23 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 27 RC Circuit A RC circuit consists of a resistor R and capacitor C. Figure 28 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 29 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 220 Current Versus Time in an RC Circuit
17 Example 29 Charging a Capacitor A circuit consists of a 26Ω resistor, a 275Ω resistor, a 82uF capacitor, a switch, and a 3.00V 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 29 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
20
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 28 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 28 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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