Electricity and Magnetism DC Circuits Resistance-Capacitance Circuits Lana Sheridan De Anza College Feb 12, 2018
Last time using Kirchhoff s laws
Overview two Kirchhoff trick problems resistance-capacitance circuits
(a) (b) (c) Using Kirchhoff s Laws examples Fig. 27-20 Question 5. 6 Res-monster maze. In Fig. 27-21, all the resistors have a resistance of 4.0 and all the (ideal) batteries have an emf of 4.0 V. What is the current through resistor R? (If you can find the proper loop through this maze, you can answer the question with a few seconds of mental calculation.) (b) Are istances o abatl. Rank through R x tion 4. Fig. 27-21 Question 6. 1 Halliday, Resnick, Walker, 8th ed, page 725, question 6.
rent i 1 through R 1 now more than, less than, or the same as previously? (c) Is the equivalent resistance R 12 of R 1 and R 2 more than, less than, or equal to R 1? Using Kirchhoff s Laws examples 8 Cap-monster maze. In Fig. 27-22, all the capacitors have a capacitance of 6.0 mf, and all the batteries have an emf of 10 V. What is the charge on capacitor C? (If you can find the proper loop through this maze, you can answer the question with a few seconds of mental calculation.) C R 2 more through current t 10 Aft closed on through gives tha values of and C 0,( 2C 0,(4) 2 with whic 11 Figu tions of nected in via a swi capacito rium) ch capacito Fig. 27-22 Question 8. 1 Halliday, Resnick, Walker, 8th ed, page 726, question 8.
Time Varying Circuits In circuits charge is not static, but moving. Current does not necessarily have to remain constant in time. Capacitors take some time to charge and discharge due to resistances in the wires. Other components also cause current to behave differently at different times, but for now, we will concentrate on circuits with resistors and capacitors.
RC Circuits Circuits with resistors and capacitors are called RC circuits. + a b S Fig. 27-15 When switch S is closed on a,the capacitor is charged through the re- R C C T cl ca ch te p p h li
Charging a Capacitor When an uncharged capacitor is first connected to an electrical potential difference, a current will flow. Once the capacitor is fully charged however, q = C ( V ), current has no where to flow and stops. The capacitor gently switches off the current.
at t 0 t Charge goes to varies with time equation acitor 16a. (27-34) e current becomes elative The capacitor s charge grows as the resistor's The charge on the capacitor current changes dies out. with time. 12 q ( µ C) i (ma) 8 4 0 2 4 6 8 10 Time (ms) 6 4 Current is the rate of charge flow with time: i = dq 2 C (a) It is possible to determine how if changes by considering the loop rule for a resistor in series with a capacitor: E /R ir q C = 0 dt.
Charge varies with time If we replace i in our equation with the derivative: E R dq dt q C = 0 This is a differential equation. There is a way to solve such equations to find solutions for how q depends on time. Here, separation of the variables q and t is possible.
Charge varies with time Rearranging: E R dq dt q C = 0 dq dt 1 CE q dq = = CE RC 1 RC dt q RC The limits of our integral will be determined by the initial conditions for the situation we are considering.
RC Circuits: Charging Capacitor When charging an initially uncharged capacitor: q = 0 at t = 0 q 0 1 CE q dq = t ln(ce q) + ln(ce 0) = t RC ( ) CE ln = t CE q RC CE CE q 0 = e t/rc 1 RC dt The solution is: q(t) = CE (1 ) e t/rc
tary RC charge.) Circuits: Note Charging Capacitor n, because at t 0 The capacitor s charge o that as t goes to q(t) = CE (1 ) grows as the resistor's o; so the equation e t/rc current dies out. n the capacitor This solution could also be in Fig. 27-16a. written in a different way. tor: C Notice Q max = CE.. (27-34) (E takes the place of the potential difference.) te that the current apacitor becomes ( ) q(t) = Q max 1 e t/rc 12 q ( µ C) 8 4 0 2 4 6 8 10 Time (ms) (a)
RC Circuits: Charging Capacitor Using the equation for q, an equation for current can also be found, since i = dq dt : i(t) = ( ) E e t/rc R Dividing the charge by the capacitance C, we can also find the potential difference across the capacitor: V C (t) = E(1 e t/rc )
cess is given in Fig. 27-16a. g the capacitor: RC Circuits: Charging Capacitor How the solutions appear with time: ing a capacitor). (27-34) The capacitor s charge Charge: grows as resistor's Current: 0 2 4 6 8 10 g. 27-16b.Note that the current q = Qcurrent max (1 dies e t/rc Time (ms) zero as the capacitor becomes out. ) i = I i e t/rc (a) 12 q ( µ C) 8 4 C 12 q ( µ C) dinary connecting C wire relative broken wire. 8 4-33, we find that the potential rging process 0 2 is 4 6 8 10 Time (ms) (a) where for this circuit ing a capacitor). (27-35) Q max = CE 6 when the capacitor /R becomes i (ma) 6 4 2 0 /R 2 4 6 8 Time (ms) 10 (b) where for this circuit I i = E R Fig. 27-16 (a) A plot of Eq. 27-33, which shows the buildup of charge on the
RC Circuits: Time Constant τ = RC τ is called the time constant of the circuit. This gives the time for the current in the circuit to fall to 1/e of its initial value. It is useful for comparing the relaxation time of different RC-circuits.
RC Circuits: Discharging Capacitor Imagine that we have charged up the capacitor, so that the charge on it is Q i. Now we flip the switch to b, the battery is disconnected, but charge flows off the capacitor, creating a current: + a b S Fig. 27-15 When switch S is closed on a,the capacitor is charged through the resistor. When the Rswitch dq is afterward closed on b,the capacitor dtdischarges q C = 0 through the resistor. R C In precedi vary with t Charging The capaci close switc capacitor, a From S charge beg terminal on plates and potential d here is equ librium (fin Here w know how across the charging p
RC Circuits: Discharging Capacitor What happens to the charge on the capacitor?
RC Circuits: Discharging Capacitor What happens to the charge on the capacitor? q(t) = Q i e t/rc It decreases exponentially with time.
RC Circuits: Discharging Capacitor R dq dt q C = 0 When discharging an initially charged capacitor: q = Q i at t = 0 The solution is: q Q i 1 q dq = t ln(q) ln(q i ) = t RC ( ) q ln = t Q i RC 0 q(t) = Q i e t/rc 1 RC dt
RC Circuits: Discharging Capacitor What happens to the current? i(t) = I i e t/rc where I i = Q i RC The negative sign means the current flows in the opposite direction through the resistor when discharging as compared with charging. i (ma) 6 4 2 /R 0 2 4 6 8 Time (ms) (b) 10
RC Circuits: Discharging Capacitor Multiplying the current by the resistance R gives the potential difference across the resistor: V R (t) = ( V ) i e t/rc The same expression describes the potential difference across the capacitor! V C (t) = ( V ) i e t/rc where ( V ) i = I i R = Q i C.
Summary resistance-capacitance circuits Next Test on Feb 15. Homework Serway & Jewett: NEW: Ch 28, onward from page 857. CQs: 7; Problems: 37, 41, 43, 45, 65, 71 NEW: Ch 26, prob: 78.