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Feb11note - PHYS142 Lecture 16 Direct Current Circuits...

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Direct Current Circuits • Chapter 28 All sections except 5 28.1 Electromotive Force 28.2 Resistors in Series and Parallel 28.3 Kirchhoff’s Rules 28.4 RC Circuits PHYS142 Lecture 16 1
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RC circuits (28.4) 1. Charging of a capacitor The uncharged capacitor is connected in series with a battery and a resistor. The switch is closed at t = 0 , and the charges start to accumulate on the plates of the capacitor. V 0 C R 2 V 0 + + _ _ C R + _ During the charging process a current (flow of charges) will exist in the circuit.
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Initial state ( t =0) Q = 0 V C = Q/C = 0 since V 0 = V R + V C V R = V 0 After a long time ( t   ) The capacitor is fully charged and no more charges can flow from the battery to the plates of the capacitor. I = 0 and V R = IR = 0 V C = V 0 V R = V 0 and the charge is maximum, Q 0 = V 0 C V 0 C R 3 V C V R
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Consider the circuit in the figure and assume the battery has no internal resistance. Just after the switch is closed, what is the current in the battery? R ε 2 / R ε / 2 R ε / 1 2 3 4 5 5% 25% 0% 28% 42% 1. 0 2. . 3. . 4. . 5. impossible to determine Question 2
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Consider the circuit in the figure and assume the battery has no internal resistance. After a very long time , what is the current in the battery? R ε 2 / R ε / 2 R ε / 1 2 3 4 5 41% 3% 0% 52% 5% 1. 0 2. . 3. . 4. . 5. impossible to determine Question 3
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To find out what the charge is at any given time, we must use Calculus . We start with the loop rule. 0 0 C R V V V 0 0 C Q IR V ) ( 1 0 0 0 CV Q RC dt dQ C Q dt dQ R V dt dQ I Using V 0 C R 6 Transient state (0 < t < ) Differential equation
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Separate the variables ( Q , t ): Integrate: Determine the constant of the integration from the initial conditions ; at t = 0, Q = 0 constant ) ln( | | ln 0 0 CV CV ) ln( | | ln 0 0 CV RC t CV Q 7 0 0 C Q dt dQ R V dt RC CV Q dQ 1 0 dt RC CV Q dQ 1 0 constant | | ln 0 RC t CV Q Charging of a capacitor
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RC t CV CV Q ) ln( | | ln 0 0 RC t CV Q RC t CV CV Q 0 0 0 1 ln ln Take exponentials : RC t e Q Q 1 0 RC t e Q Q CV Q 0 0 1 1 8 Charging of a capacitor
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The quantity RC has units of time. It is called the characteristic time or the time constant for charging: = RC τ
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Feb11note - PHYS142 Lecture 16 Direct Current Circuits...

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