1Bweek9. - Physics 1 B Class Notes Walter Gekelman Spring...

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1 Physics 1 B Class Notes© Spring Quarter 2010 Walter Gekelman DC Circuits – Week 9 Consider an RC circuit containing a capacitor a resistor and a battery. What happens when you close the switch? E R S C Assume the current is counterclockwise and use Kirchoff’s loop law: (1) ε iR q C = 0 ; i = dq dt ; R dq dt q C = 0 Assume the solution is of the form q = Ae α t + B Then substituting: = AR ( ) e t + A C e t + B To satisfy this equation we must equate the constant and exponential terms separately = B ; AR ( ) e t + A C e t = 0 ; = 1 RC Before we close the switch at t = 0- there is no charge on the capacitor , q(0-)=0 therefre B=-A. When the capacitor charges up a long time after the switch is closed the current is zero and the charge on the capacitor is q ( ) = C . Therefore q = C 1 e t RC . RC is the “time constant” of the circuit. It has the dimension of time. Here is the q(t) of a circuit where R = 100 Ohms and C = 1 mF. T is in seconds
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2 The current is i = dq dt = ε R e t RC i (amps) t (seconds) In 1 RC time constant it drops to 1/e or about 1/3 of its initial value. Note at t=0+ the current is the same as if the capacitor is a short circuit. A capacitor behaves as a short circuit when it begins charging up and an open circuit one it is fully charged. Various problems: Consider the circuit below. Find the current in all the loops:
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3 leftloop ; 36-6i 1 3 i 1 i 2 ( ) = 0 bottomloop ; -3i 2 6 ( I i 1 + i 2 ( ) + 3 i 1 i 2 ( ) = 0 We the get I = 12A, i 1
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1Bweek9. - Physics 1 B Class Notes Walter Gekelman Spring...

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