Electricity and Magnetism Review 2: Units 7-11 Mechanics Review 2, Slide 1

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

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 ?

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.

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

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

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

 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

 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.

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

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.

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 )

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 x04x04 C and Q change, V stays the same.

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 34  14 C  C0 34  14  What is Q ? Q f  VC 0  3  4  1  4 

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:

Example: Circuits (Without Kirchhoff’s Rules) R 3 and R 24 are connected in parallel  R  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

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 )?

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

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

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 = Amps I 3 = 1.02 Amps Q = 66.0 μC V gd = 2.90 V