1. Microscopic treatment: insight into the fundamental physical mechanism of circuit behavior. Not easy to measure directly E, u, Q, v. It is easier to measure conventional current, potential difference  macroscopic parameters Need a link between microscopic and macroscopic quantities. Macroscopic Analysis of Circuits

2. Many elements in a circuit act as resistors: prevent current from rising above a certain value. Goal: find a simple parameter which can predict  V and I in such elements. Need to combine the properties of material and geometry. Resistance

3. Conventional current: Different properties of the material Geometry Group the material properties together: Current density: Conductivity Combining the properties of a material

4. In copper at room temperature, the mobility of electrons is about 4.5. 10 -3 (m/s)/(V/m) and the density of electrons is n=8. 10 28 m -3. What is  ? What is the strength of E required to drive a current of 0.3 A through a copper wire which has a cross-section of 1 mm 2 ? Exercise

5. The conductivity of tungsten at RT is  =1.8. 10 7 (A/m 2 )/(V/m) and it decreases 18 times at a temperature of a glowing filament (3000 K). The tungsten filament has a radius of 0.015 mm. What is E required to dive 0.3A through it? Exercise

6. Conductivity with two Kinds of Charge Carriers

7. Conventional current: Widely known as Ohm’s law Resistance of a long wire: Units: Ohm,  George Ohm (1789-1854) Resistance Resistance combines conductivity and geometry!

8. Microscopic Macroscopic Can we write V=IR ? Microscopic and Macroscopic View Current flows in response to a  V

9. L=5 mm A = 0.002 mm 2 Conductivity of Carbon:  = 3. 10 4 (A/m 2 )/(V/m) What is its resistance R? (V/A) What would be the current through this resistor if connected to a 1.5 V battery? Exercise: Carbon Resistor

10. Mobility of electrons: depends on temperature Conductivity and resistance depend on temperature. Constant and Varying Conductivity

11. Ohmic resistor: resistor made of ohmic material… Ohmic materials: materials in which conductivity  is independent of the amount of current flowing through not a function of current Examples of ohmic materials: metal, carbon (at constant T!) Ohmic Resistors

12. Tungsten: mobility at room temperature is larger than at ‘glowing’ temperature (~3000 K) V-A dependence: 3 V100 mA 1.5 V 80 mA 0.05 V 6 mA R 30  19  8  VV I Is a Light Bulb an Ohmic Resistor? Clearly not ohmic!

13. Metals, mobile electrons: slightest  V produces current. If electrons were bound – we would need to apply some field to free some of them in order for current to flow. Metals do not behave like this! Semiconductors: n depends exponentially on E Conductivity depends exponentially on E Conductivity rises (resistance drops) with rising temperature Semiconductors

14. Capacitors |  V|=Q/C, function of time Batteries: double current, but |  V|  emf, hardly changes has limited validity! Ohmic when R is indep- pendent of I! Conventional symbols: Nonohmic Circuit Elements Semiconductors

15.  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 Series Resistance

16. A certain ohmic resistor has a resistance of 40 . A second resistor is made of the same material, but is three times longer and has a half of the cross-sectional area. What is its resistance? Resistor 1:Resistor 2: What would be an equivalent resistance of these two resistors in series? Exercise

17. Know R, find  V 1,2 Solution: 1) Find current: 2) Find voltage: 3) Check: Exercise: Voltage Divider R1R1 R2R2 V1V1 V2V2 emf

18. I = I 1 + I 2 + I 3 Parallel Resistance

19. R 1 = 30  R 2 = 10  What is the equivalent resistance? What is the total current? Alternative way: Two Light Bulbs in Parallel

20. What would you expect if one is unscrewed? Two Light Bulbs in Parallel A)The single bulb is brighter B)No difference C)The single bulb is dimmer

21. Current: charges are moving  work is done Work = change in electric potential energy of charges Power = work per unit time: I Power for any kind of circuit component: Work and Power in a Circuit Units: W =

22. I = 0.3 A Units: Example: Power of a Light Bulb emf = 3V

23. emf R Know  V, find P Know I, find P In practice: need to know P to select right size resistor – capable of dissipating thermal energy created by current. Power Dissipated by a Resistor

24. Alternative approach: Energy density: Energy: Energy Stored in a Capacitor

25. Model of a real battery ideal battery R Round trip (energy conservation): r int  0.25  R 100  10  1  0  Ideal 0.015 A 0.15 A 1.5 A infinite 1.5 V Real 0.01496 A 0.146 A 1.2 A 6 A  V R =RI 1.496 V 1.46 V 1.2 V 0 V Real Batteries: Internal Resistance