1. The parallel-plate capacitor charged with q, then: energy density

2. Dielectrics and Gauss’ Law A dielectric slab is inserted, q ’ is induced surface charge. k e   instead of  . k e   instead of  . The charge q contained within the Gauss surface is taken to be the free charge only. is taken to be the free charge only. Gauss’ law should be amended as:

3. Dielectrics and Gauss’ Law Electric displacement vector Electric polarization vector

4. Chapter 31 DC Circuits

5. Circuit s Direct Circuits (DC) Alternating Circuits (AC) Resistor Battery Current The direction of current is the direction of positive charge would move. V(t)

6. Resistor Battery Current The direction of current is the direction of positive charge would move. i pump for charge A “pump” for charge, maintains the constant potential difference between its two terminals. A source of energy to raise the energy of electrons.

7. Conservation of Charge Consider the flux of current density through a closed surface, Therefore the is expressed as: Therefore the law of charge conservation is expressed as: j j For a steady current (e.g., the current in DC circuit),

8. Conservation of Charge Consider the flux of current density through a closed surface, j j Consequently, at any junction in an electric circuit, A1A1A1A1 A3A3A3A3 A2A2A2A2 the total current entering the junction must be equal to the total current leaving the junction. i1i1 i2i2 i3i3

9. i1i1i1i1 i2i2i2i2 i3i3i3i3 iNiNiNiN In general, This is called junction rule (Kirchhoff’s first law).

10. A device that maintains a constant potential difference between two points in the circuit. Electromotive Force(EMF) + - The EMF  of a source is defined as the work on per unit positive charge, Joule/Coulomb = Volt Does this by moving charges from low to high potential by doing work. The EMF is a device transferring from variety of energy to electric energy. Uses chemical energy to do work Battery : Uses chemical energy to do work Uses mechanical energy to do work Generator : Uses mechanical energy to do work Uses light to do work Solar Cell : Uses light to do work

11. Analysis of Circuits i +  V High V Low - Method of potential differences : differences in potential across each circuit element. Guess a direction for the current first. Passing through a resistor in direction of current Passing through a resistor in direction of current flow, from a high potential to a low potential gives flow, from a high potential to a low potential gives  V = V final - V initial = V low - V high = - iR (<0) Passing through battery from “- ” to “+”, the potential increases so Passing through battery from “- ” to “+”, the potential increases so  V = V high - V low =  (>0) Making a complete loop gives total  V =0 !!! This is called loop rule (Kiechhoff’s second law)

12. It can be used the following two rules: 1. Junction Rule At any junction in an electric circuit, the total current entering the junction must be the same as the total current leaving the junction. 2. Loop Rule The algebraic sum of all differences in potential around a complete circuit loop must be zero. What is “±” for the differences potential of resistor? What is “±” for the differences potential of EMF?

13. Analysis of Circuits i +  - R Real batteries have internal resistance. So what does our circuit really look like?  r

14. Multi-loop DC Circuits Examples: It can be reduced to a simple one-loop circuit.

15. Example Find i 1, i 2, i 3. i1i1 i3i3 i2i2 The junction rule at junction B gives: The loop rule leads to: – –There is another loop (around outside) but it gives no new information, just the sum of the equations above:

16. Example Find i 1, i 2, i 3. i1i1 i3i3 i2i2 The junction rule at junction B gives: The loop rule leads to:

17. Example Find i 1, i 2, i 3. i1i1 i3i3 i2i2 The junction rule at junction B gives: The loop rule leads to: Note that i 1 and i 3 turned out negative!This means those two currents are flowing opposite to the directions assumed

18. Example Find i 1, i 2, i 3. i1i1 i3i3 i2i2 The junction rule at junction B gives: Matrix methodRI=V I=VR -1

19. Electric Fields in Circuits Where does the electric field in wires come from? In a conductor, A tiny amounts of charge on the surface of wires provide the electric field

20. Energy Transfers in an Electric Circuit i As the battery moves a quantity of charge dq from its negative terminal to its positive terminal, it does work The power delivered by the source of EMF (battery) is then: The potential difference between two terminals of the resistor is, As a dq moves through the resistor, it experiences a potential energy change: This energy must be transferred to the resistor, known as Joule Heating. The power transferred to the resistor reads:

21. The charge dq passing through the battery gains potential energy : The power delivered by this battery is: In a real battery with internal resistance r, the potential difference between the terminals is,

22. RC Circuits Combine Resistor and Capacitor in Series C  a R b Switch at position (a) V C (t) =? At t = 0, q(0) = 0 At t= , q(  ) = 0.63C  At t = , q(  ) = C  At t = 0, i(0) =  / R At t= , i(  ) = 0. 37  R At t = , i(  ) = 0 At t = 0, V c (0) = 0 At t= , V c (  ) = 0.63  At t = , V c (  ) = 

23. Then switch turn to b, C discharges: C  a R b

24. Time to charge capacitor  a R b C 12  1F1F How long time dose voltage of C to  /2 ? Voltage on capacitor: Example

25. R C  a b 12  1F1F How long time dose voltage of C reduce to  /2 ? Voltage on capacitor: Example

26. How long time dose voltage of C reduce to  /2 ? R C  ab 12  1F1F Example

27. Energy? R C  ab 12  1F1F Example

30. Exercises P719~722 11, 13, 25, 47 Problems P724 15

31. Example