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Electricity and Magnetism Review 2: Units 7-11 Mechanics Review 2, Slide 1
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R3R3 V R1R1 R2R2 C S Immediately after S is closed: what is V C, the voltage across C ? Example: RC Circuit In this circuit, assume V, C, and R i are known. C is initially uncharged and then switch S is closed. What is the voltage across the capacitor after a very long time ? V C 0 what is I 2, the current through R 2 ? At t = 0 the capacitor behaves like a wire. Solve using Kirchhoff’s Rules. V R1R1 R2R2 S R3R3
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V R1R1 R2R2 C R3R3 S Immediately after S is closed, what is I 1, the current through R 1 ? V R1R1 R2R2 S R3R3 V C 0 I1I1 Example: RC Circuit In this circuit, assume V, C, and R i are known. C is initially uncharged and then switch S is closed. What is the voltage across the capacitor after a very long time ?
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Example: RC Circuit V R1R1 R2R2 C R3R3 S After S has been closed “for a long time”, what is I 2, the current towards C ? V R1R1 R3R3 I 2 0 VCVC I In this circuit, assume V, C, and R i are known. C is initially uncharged and then switch S is closed. What is the voltage across the capacitor after a very long time ? After a long time the capacitor and R 2 are are not connected to the circuit.
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V R1R1 R2R2 C R3R3 S V C V 3 IR 3 (V/(R 1 R 3 ))R 3 V R1R1 R3R3 VCVC I I In this circuit, assume V, C, and R i are known. C is initially uncharged and then switch S is closed. What is the voltage across the capacitor after a very long time ? Example: RC Circuit
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In this circuit, assume V, C, and R i are known. C initially uncharged and then switch S is closed for a very long time charging the capacitor. Then the switch is opened at t = 0. Redraw the circuit with the switch open. Now it looks like a simple RC circuit. What is disc, the discharging time constant? What is the current on R 3 as a function of time? R2R2 C R3R3 Example RC Circuit V R1R1 R2R2 C R3R3 S
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Example: Capacitors Three capacitors are connected to a battery as shown. A) What is the equivalent capacitance? B) What is the total charge stored in the system? C) Find the charges on each capacitor. V Parallel: C 23 C 2 C 3 Series: 1 C 123 1 C 23 1 C 1 Total Charge: Q = C 123 V Charges on capacitors: Q 1 = Q Q 2 + Q 3 = Q V 2 = V 3
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Batteries are easy V1V1 R1R1 R2R2 In this circuit V i and R i are known. What are the currents I 1, I 2, I 3 ? R3R3 V2V2 V3V3 I1I1 I3I3 I2I2 Label and pick directions for each current Label the + and - side of each element For resistors, the “upstream” side is + Example: Kirchhoff’s Rules
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V1V1 R1R1 R2R2 In this circuit V i and R i are known. What are the currents I 1, I 2, I 3 ? R3R3 V2V2 V3V3 I1I1 I3I3 I2I2 Example: Kirchhoff’s Rules 1. I 2 I 1 I 3 2. V 1 I 1 R 1 I 3 R 3 V 3 = 0 3. V 3 I 3 R 3 I 2 R 2 V 2 = 0 4. V 2 I 2 R 2 I 1 R 1 V 1 = 0 We need 3 equations: Which 3 should we use? Kirchhoff’s Rules give us the following 4 equations : The node equation (1.) and any two loops.
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First determine E field produced by charged conductors: Integrate E to find the potential difference V Example: Calculating Capacitance A solid cylindrical conductor of radius a length l and charge Q is coaxial with a thin cylindrical shell of radius b and charge –Q. Assume l is much larger than b. Find the capacitance
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First determine E field produced by charged conductors: Integrate E to find the potential difference V Example: Calculating Capacitance A spherical conducting shell of radius b and charge –Q is concentric with a smaller conducting sphere of radius a and charge Q. Find the capacitance.
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Example: Capacitors In the circuit shown the switch S A is originally closed and the switch S B is open. (a) What is the initial charge on each capacitor. Then S A is opened and S B is closed. (b) What is the final charge on each capacitor. (c) Now S A is closed also. How much additional charge flows though S A ? Initial Charge: Q 1i = C 1 ΔV Q 2i = Q 3i = 0 Final Charge: Q 1f = C 1 ΔV f Q 2f = Q 3f ΔV f = Q 2f /C 2 + Q 3f /C 3 Q 1i = Q 1f + Q 2f Both switches closed: Q total = C total ΔV C total = C 1 +1/(1/C 2 +1/C 3 )
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Example: Capacitor with Dielectric An air-gap capacitor, having capacitance C 0 and width x 0 is connected to a battery of voltage V. A dielectric ( ) of width x 0 /4 is inserted into the gap as shown. What is Q f, the final charge on the capacitor? What changes when the dielectric added? V C0C0 x0x0 V x04x04 C and Q change, V stays the same.
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Example: Capacitor with Dielectric Can consider capacitor to be two capacitances, C 1 and C 2, in parallel C 1 3 / 4 C 0 What is C 1 ? For parallel plate capacitor: C 0 A/d C1C1 C2C2 What is C 2 ? C2 1/4 C0C2 1/4 C0 What is C ? C C0 34 14 C C0 34 14 What is Q ? Q f VC 0 3 4 1 4
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Example: Circuits Redraw the circuit using the equivalent resistor R 24 series combination of R 2 and R 4. R 2 and R 4 are connected in series R 24 which is connected in parallel with R 3 V R1R1 R2R2 R4R4 R3R3 In the circuit shown: V 18V, R 1 1 R 2 2 R 3 3 and R 4 4 What is V 2, the voltage across R 2 ? V R1R1 R3R3 R 24 R 24 R 2 R 4 2 4 6 First combine resistors to find the total current:
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Example: Circuits (Without Kirchhoff’s Rules) R 3 and R 24 are connected in parallel R 234 1 R eq 1 R a 1 R b In the circuit shown: V 18V, R 1 1 R 2 2 R 3 3 and R 4 4 What is V 2, the voltage across R 2 ? V R1R1 R3R3 R 24 1 R 234 1 3 1 6 3 6 -1 2 V R1R1 R 234 R 1 and R 234 are in series. R 1234 1 2 3 I 1234 V R 1234 Ohm’s Law I 1 I 1234 V R 1234 6 Amps
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a b Example: Circuits R 234 Since R 1 in series with R 234 I 234 I 1234 I 1 6 Amps V 234 I 234 R 234 6 x 2 12 Volts In the circuit shown: V 18V, R 1 1 R 2 2 R 3 3 and R 4 4 R 24 6 R 234 2 I 1234 6 A What is V 2, the voltage across R 2 ? I 1234 V R 1234 V R1R1 R 234 I 1 I 234 What is V 234 (V ab )?
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Example: Circuits V R1R1 R 24 R3R3 I 24 I 2 2Amps V R1R1 R2R2 R4R4 R3R3 I 1234 Ohm’s Law V 2 I 2 R 2 4 Volts I 24 I 24 V 234 / R 24 2Amps
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Example: Circuits What is I 3 ? V R1R1 R 234 a b V R1R1 R2R2 R4R4 R3R3 V 18V R 1 1 R 2 2 R 3 3 R 4 4 R 24 6 R 234 2 V 234 = 12V V 2 = 4V I 1 = 6 Amps I 2 = 2 Amps I1I1 I2I2 I3I3 I1 I2 I3I1 I2 I3 I 3 4 A What is P 2 ? P 2 I 2 V 2 I 2 2 R 2 W
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Example: Kirchhoff’s Rules Given the circuit below. Use Kirchhoff’s rules to find the currents I 1, I 2 and I 3, and the charge Q on the capacitor. What is the voltage difference between points g and d? (Assume that the circuit has reached steady state currents) I 1 = 1.38 Amps I 2 = - 0.364 Amps I 3 = 1.02 Amps Q = 66.0 μC V gd = 2.90 V
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