1. Tues. Oct. 6, 2009Physics 208 Lecture 101 Last time… Fields, forces, work, and potential Electric forces and work + + Potential energy stored in electric field

2. Tues. Oct. 6, 20092 Work, KE, and potential energy When particle is not isolated, Work done on system Change in kinetic energy Change in electric potential energy Works for constant electric field if Only electric potential energy difference Sometimes a reference point is chosen E.g. Then for uniform electric field

3. Tues. Oct. 6, 20093 Electric potential V Electric potential difference  V is the electric potential energy / unit charge =  U/q For uniform electric field, This is only valid for a uniform electric field

4. Tues. Oct. 6, 20094 Check for uniform E-field + + Push particle against E-field, or across E-field Which requires work? Constant electric potential in this direction Decreasing electric potential in this direction Increasing electric potential in this direction

5. Tues. Oct. 6, 20095 Quick Quiz Two points in space A and B have electric potential V A =20 volts and V B =100 volts. How much work does it take to move a +100µC charge from A to B? A.+2 mJ B.-20 mJ C.+8 mJ D.+100 mJ E.-100 mJ

6. Tues. Oct. 6, 2009Physics 208 Lecture 106 Potential from electric field dV largest in direction of E-field. dV smallest (zero) perpendicular to E-field V=VoV=Vo

7. Tues. Oct. 6, 20097 Electric potential: general Electric field usually created by some charge distribution. V(r) is electric potential of that charge distribution V has units of Joules / Coulomb = Volts Electric potential energy difference  U proportional to charge q that work is done on Electric potential difference Depends only on charges that create E-fields

8. Tues. Oct. 6, 20098 Electric potential of point charge Electric field from point charge Q is What is the electric potential difference? Define for point charge Then

9. Tues. Oct. 6, 2009Physics 208 Lecture 109 Equipotential lines Lines of constant potential In 3D, surfaces of constant potential

10. Tues. Oct. 6, 2009Physics 208 Lecture 1010 Topographic map Each lines is constant elevation Same as constant gravitational potential gh (energy = mgh) Height interval between lines constant

11. Tues. Oct. 6, 2009Physics 208 Lecture 1011 Electric field from potential Spell out the vectors: Usually written Said before that This works for

12. Tues. Oct. 6, 2009Physics 208 Lecture 1012 Quick Quiz Suppose the electric potential is constant everywhere. What is the electric field? A)Positive B)Negative C)Increasing D)Decreasing E)Zero

13. Tues. Oct. 6, 2009Physics 208 Lecture 1013 Electric Potential - Uniform Field Constant E-field corresponds to linearly decreasing (in direction of E) potential A B x Here V depends only on x, not on y

14. Tues. Oct. 6, 2009Physics 208 Lecture 1014 Check of basic cases Previous quick quiz: uniform potential corresponds to zero electric field Linear potential corresponds to constant electric field

15. Tues. Oct. 6, 2009Physics 208 Lecture 1015 Potential and charge Have shown that a conductor has an electric potential, and that potential depends on its charge For a charged conducting sphere: + + + + + + + + + + + Electric potential proportional to total charge

16. Tues. Oct. 6, 2009Physics 208 Lecture 1016 Quick Quiz Consider this conducting object. When it has total charge Q o, its electric potential is V o. When it has charge 2Q o, its electric potential A. is V o B. is 2V o C. is 4V o D. depends on shape

17. Tues. Oct. 6, 2009Physics 208 Lecture 1017 Capacitance Electric potential of any conducting object proportional to its total charge. C = capacitance Large capacitance: need lots of charge to change potential Small capacitance: small charge can change potential.

18. Tues. Oct. 6, 2009Physics 208 Lecture 1018 Capacitors Where did the charge come from? Usually transferred from another conducting object, leaving opposite charge behind A capacitor consists of two conductors Conductors generically called ‘plates’ Charge transferred between plates Plates carry equal and opposite charges Potential difference between plates proportional to charge transferred Q

19. Tues. Oct. 6, 2009Physics 208 Lecture 1019 Definition of Capacitance Same as for single conductor but  V = potential difference between plates Q = charge transferred between plates SI unit of capacitance is farad (F) = 1 Coulomb / Volt This is a very large unit: typically use   F = 10 -6 F, nF = 10 -9 F, pF = 10 -12 F

20. Tues. Oct. 6, 2009Physics 208 Lecture 1020 How was charge transferred? Battery has fixed electric potential difference across its terminals Conducting plates connected to battery terminals by conducting wires.  V plates =  V battery across plates Electrons move from negative battery terminal to -Q plate from +Q plate to positive battery terminal This charge motion requires work The battery supplies the work VV

21. Tues. Oct. 6, 2009Physics 208 Lecture 1021 Requires work to transfer charge dq from one plate: Work done to charge a capacitor Work done stored as potential energy in capacitor Total work = sum of incremental work

22. Tues. Oct. 6, 2009Physics 208 Lecture 1022 Example: Parallel plate capacitor Charge Q moved from right conductor to left conductor Charge only on inner surfaces Plate surfaces are charge sheets, each producing E-field +Q -Q d inner outer Uniform field between plates

23. Tues. Oct. 6, 2009Physics 208 Lecture 1023 Quick Quiz Electric field between plates of infinite parallel-plate capacitor has a constant value  /  o. What is the field outside of the plates? A.  /  o B.  /2  o C.-  /2  o D.  /4  o E.0

24. Tues. Oct. 6, 2009Physics 208 Lecture 1024 What is potential difference? -Q +Q d + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - Potential difference = V + -V - = - (work to move charge q from + plate to - plate) / q

25. Tues. Oct. 6, 2009Physics 208 Lecture 1025 What is the capacitance? +Q+Q -Q-Q d This is a geometrical factor Energy stored in parallel-plate capacitor Energy density

26. Tues. Oct. 6, 2009Physics 208 Lecture 1026 Human capacitors Cell membrane: ‘Empty space’ separating charged fluids (conductors) ~ 7 - 8 nm thick In combination w/fluids, acts as parallel-plate capacitor Cytoplasm Extracellular fluid Plasma membrane 100 µm

27. Tues. Oct. 6, 2009Physics 208 Lecture 1027 Modeling a cell membrane Charges are +/- ions instead of electrons Charge motion is through cell membrane (ion channels) rather than through wire Otherwise, acts as a capacitor ~0.1 V ‘resting’ potential Cytoplasm Extracellular fluid Plasma membrane Na + Cl - K+K+ A-A- - ----- + +++++ 7-8 nm  V~0.1 V Ionic charge at surfaces of conducting fluids Capacitance: 100 µm sphere surface area ~ 3x10 -4 cm 2 ~0.1µF/cm 2

28. Tues. Oct. 6, 2009Physics 208 Lecture 1028 Cell membrane depolarization Cell membrane can reverse potential by opening ion channels. Potential change ~ 0.12 V Ions flow through ion channels Channel spacing ~ 10xmembrane thickness (~ 100 channels / µm 2 ) How many ions flow through each channel? Cytoplasm Extracellular fluid Plasma membrane Na + Cl - K+K+ A-A- - ----- + +++++ 7-8 nm  V~0.1 V + +++++ - -----  V~-0.02 V (100 channels/µm 2 )x4  (50 µm) 2 =3.14x10 6 ion channels Charge xfer required  Q=C  V=(35 pF)(0.12V)=(35x10 -12 C/V)(0.12V) = 4.2x10 -12 Coulombs 1.6x10 -19 C/ion -> 2.6x10 7 ions flow Ion flow / channel =(2.6x10 7 ions) / 3.14x10 6 channels ~ 7 ions/channel

29. Tues. Oct. 6, 2009Physics 208 Lecture 1029 Cell membrane as dielectric Membrane is not really empty It has molecules inside that respond to electric field. The molecules in the membrane can be polarized Cytoplasm Extracellular fluid Plasma membrane Na + Cl - K+K+ A-A- - ----- + +++++ 7-8 nm Dielectric: insulating materials can respond to an electric field by generating an opposing field.

30. Tues. Oct. 6, 2009Physics 208 Lecture 1030 Effect of E-field on insulators If the molecules of the dielectric are non-polar molecules, the electric field produces some charge separation This produces an induced dipole moment + - + - E=0 E

31. Tues. Oct. 6, 2009Physics 208 Lecture 1031 Dielectrics in a capacitor An external field can polarize the dielectric The induced electric field is opposite to the original field The total field and the potential are lower than w/o dielectric E = E 0 /  and  V = V 0 /  The capacitance increases C =  C 0 E0E0 E ind

32. Tues. Oct. 6, 2009Physics 208 Lecture 1032 Cell membrane as dielectric Without dielectric, we found 7 ions/channel were needed to depolarize the membrane. Suppose lipid bilayer has dielectric constant of 10. How may ions / channel needed? Cytoplasm Extracellular fluid Plasma membrane Na + Cl - K+K+ A-A- - ----- + +++++ 7-8 nm C increases by factor of 10 10 times as much charged needed to reach potential A.70 B.7 C.0.7