EXAM2  Wednesday, March 26, 8:00-10:00 pm  No lecture on that day.  room 112 for students in R21/22/23/24  room 114 for students in R25/26/27  Chapters 17, 18, 19 & 20  Special needs:  Students have been contacted for special arrangement. Please let me know if you haven’t receive the .  AOB – multiple choice. – Prepare your own scratch paper, pens, pencils, erasers, etc. Use only pencil for the answer sheet Bring your own calculators No cell phones, no text messaging which is considered cheating. No crib sheet of any kind is allowed. Equation sheet will be provided. 1

Last time Different configuration of circuit Charge and discharge capacitors Different way of connecting capacitors. (Capacitance) (More details on resistance) 2

Today Capacitance Different way of connecting resistors Solving complicated circuit using Kirchhoff’s Rules (Ammeters, Voltmeters and Ohmmeters) 3

Electric field in a capacitor: E s +Q -Q In general: Definition of capacitance: Capacitance Capacitance of a parallel- plate capacitor: Capacitance Wiki: The SI unit of capacitance is the farad (symbol: F), named after the English physicist Michael FaradaySIfarad Michael Faraday 4

A capacitor is formed by two rectangular plates 50 cm by 30 cm, and the gap between the plates is 0.25 mm. What is its capacitance? Exercise 5

s D No insulator:With insulator: A Capacitor With an Insulator Between the Plates 6

Unit: J, A/m 2 ; E, V/m; , (A/m 2 )/(V/m) 7

Conductivity with two kinds of charge carrier 8

Resistance is in the unit of Ohm. 9

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Kirchhoff’s Rules Kirchhoff’s Rule 2: Loop Rule  When any closed loop is traversed completely in a circuit, the algebraic sum of the changes in potential is equal to zero. Kirchhoff’s Rule 1: Junction Rule  The sum of currents entering any junction in a circuit is equal to the sum of currents leaving that junction.  Conservation of charge  In and Out branches  Assign I i to each branch  Coulomb force is conservative 11

 V batt +  V 1 +  V 2 +  V 3 = 0 emf - R 1 I - R 2 I - R 3 I = 0 emf = R 1 I + R 2 I + R 3 I emf = (R 1 + R 2 + R 3 ) I emf = R equivalent I, where R equivalent = R 1 + R 2 + R 3 For resistors made of the same material and with the same A it follows straight from the definition of resistance: Series Resistance 12

I = I 1 + I 2 + I 3 For resistors made of the same material and with the same A it follows straight from the definition resistance: Parallel Resistance 13

Drift speed of ions in chemical battery: In usual circuit elements: In a battery:, assuming uniform field: emf r int - internal resistance Real Batteries: Internal Resistance 14

Circuit Analysis Tips Simplify using equivalent resistors Label currents with arbitary directions If the calculated current is negative, the real direction is opposite to the one defined by you. Apply Junction Rule to all the labeled currents. Useful when having multiple loops in a circuit. Choose independent loops and define loop direction Imagine your following the loop and it’s direction to walk around the circuit. Use Loop Rule for each single loop If current I direction across a resistor R is the same as the loop direction, potential drop across R is ∆V = −I×R, otherwise, ∆V = I×R For a device, e.g. battery or capacitor, rely on the direction of the electric field in the device and the loop direction to determine the Potential drop across the device Solve simultaneous linear equations 15

Loop Example with Two EMF Devices  If  1 <  2, we have I<0 !? This just means the actual current flows reverse to the assumed direction. No problem! 16

Finding Potential and Power in a Circuit Just means 0 V here But what is I? Must solve for I first! supplied by 12V battery dissipated by resistors The rest? into 4V battery (charging) 17

Charging a Battery Positive terminal to positive terminal Charging EMF > EMF of charged device Say, R+r 1 +r 2 =0.05  (R is for jumper cables). Then, power into battery 2 battery being charged (11V) good battery (12V) If connected backward,  Large amount of gas produced  Huge power dissipation in wires 18

Using Kirchhoff’s Laws in Multiple Loop Circuits Identify nodes and use Junction Rule: Identify independent loops and use Loop Rule: Only two are independent. 19

Example What’s the current I 1 ? I 1 +I 2 I2I2 I1I1 (a). 2.0A (b). 1.0A (c). -2.0A (d). -1.0A (e). Need more information to calculate the value. 20

Replace by equivalent R=2  first. Sketch the diagram Simplify using equivalent resistors Label currents with directions Use Junction Rule in labeling Choose independent loops Use Loop Rule Solve simultaneous linear equations I 1 +I 2 I2I2 I1I1 21

What is the current through R 1 ? a.0.575A b.0.5A c.0.75A d.0.33A e.1.5A iClicker Question 45V 30  45V 30  R1R1 R2R2 R3R3 22

Ammeter: measures current I Voltmeter: measures voltage difference  V Ohmmeter: measures resistance R Ammeters, Voltmeters and Ohmmeters 23

Ammeter is inserted in series into a circuit – measured current flows through it. Remf A Process of measuring requires charges to do some work: Internal resistance r int A No ammeter: With ammeter: Internal resistance of an ammeter must be very small Ammeter Design: r int 24

 V AB – add a series resistor to ammeter Measure I and convert to  V AB =IR Connecting Voltmeter: Higher potential must be connected to the ‘+’ socket and lower one to the ‘-’ socket to result in positive reading. Voltmeter Voltmeters measure potential difference 25

R1R1 R2R2 emf A B  V AB in absence of a voltmeter A r int  V AB in presence of a voltmeter Internal resistance of a voltmeter must be very large Voltmeter: Internal Resistance 26

R How would you measure R? A Ohmmeter Ammeter with a small voltage source 27

Initial situation: Q=0 Q and I are changing in time Quantitative Analysis of an RC Circuit 28

Current in an RC circuit What is I 0 ? Current in an RC circuit RC Circuit: Current 29

Current in an RC circuit What about charge Q? Current in an RC circuit RC Circuit: Charge and Voltage Check: t=0, Q=0, t--> inf, Q=C*emf 30

Current in an RC circuit Charge in an RC circuit Voltage in an RC circuit RC Circuit: Summary 31

Current in an RC circuit When time t = RC, the current I drops by a factor of e. RC is the ‘time constant’ of an RC circuit. The RC Time Constant A rough measurement of how long it takes to reach final equilibrium 32