Last time… Fields, forces, work, and potential

+ + Last time… Fields, forces, work, and potential Electric forces and work Potential energy stored in electric field Physics 208 Lecture 10

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

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

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

Quick Quiz Two points in space A and B have electric potential VA=20 volts and VB=100 volts. How much work does it take to move a +100µC charge from A to B? +2 mJ -20 mJ +8 mJ +100 mJ -100 mJ

Potential from electric field dV largest in direction of E-field. dV smallest (zero) perpendicular to E-field V=Vo Physics 208 Lecture 10

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

for point charge Electric potential of point charge • Electric field from point charge Q is • What is the electric potential difference? Define Then

Equipotential lines • Lines of constant potential • In 3D, surfaces of constant potential Physics 208 Lecture 10

Topographic map • Each lines is constant elevation • Same as constant gravitational potentialgh (energy = mgh) • Height interval between lines constant Physics 208 Lecture 10

Electric field from potential • Said before that • Spell out the vectors: • This works for Usually written Physics 208 Lecture 10

Quick Quiz Suppose the electric potential is constant everywhere. What is the electric field? Positive Negative Increasing Decreasing Zero Physics 208 Lecture 10

Electric Potential - Uniform Field B Constant E-field corresponds to linearly decreasing (in direction of E) potential A Here V depends only on x, not on y x Physics 208 Lecture 10

Check of basic cases • Previous quick quiz: uniform potential corresponds to zero electric field • Linear potential corresponds to constant electric field Physics 208 Lecture 10

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 + + + + Physics 208 Lecture 10

Quick Quiz Consider this conducting object. When it has total charge Qo, its electric potential is Vo. When it has charge 2Qo, its electric potential A. is Vo B. is 2Vo C. is 4Vo D. depends on shape Physics 208 Lecture 10

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. Physics 208 Lecture 10

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 platesproportional to charge transferred Q Physics 208 Lecture 10

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 • mF = 10-6 F, nF = 10-9 F, pF = 10-12 F Physics 208 Lecture 10

How was charge transferred? DV • Battery has fixed electric potential difference across its terminals • Conducting plates connected to battery terminals by conducting wires. • DVplates = DVbattery 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 Physics 208 Lecture 10

Work done to charge a capacitor • Requires work to transfer charge dq from one plate: • Total work = sum of incremental work • Work done stored as potential energy in capacitor Physics 208 Lecture 10

+Q -Q outer inner d 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 Uniform field between plates Physics 208 Lecture 10

Quick Quiz Electric field between plates of infinite parallel-plate capacitor has a constant value /o. What is the field outside of the plates? /o /2o - /2o /4o 0 Physics 208 Lecture 10

- + - + - + - + - + + - - + - + + - + - + - + - - + - + - + What is potential difference? Potential difference = V+-V- = - (work to movecharge qfrom + plate to - plate) / q d -Q +Q Physics 208 Lecture 10

What is the capacitance? -Q This is a geometrical factor +Q Energy stored in parallel-plate capacitor Energy density d Physics 208 Lecture 10

Extracellular fluid Plasma membrane Cytoplasm Human capacitors • Cell membrane: • ‘Empty space’ separating charged fluids (conductors) • ~ 7 - 8 nm thick • In combination w/fluids, acts as parallel-plate capacitor 100 µm Physics 208 Lecture 10

A- K+ Extracellular fluid 7-8 nm V~0.1 V Plasma membrane - - - - - - Cytoplasm Na+ Cl- + + + + + + 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 Ionic charge at surfaces of conducting fluids ~ 3x10-4 cm2 100 µm sphere surface area Capacitance: ~0.1µF/cm2 Physics 208 Lecture 10

- - - - - - - - - - - - + + + + + + + + + + + + Cell membrane depolarization A- K+ Extracellular fluid • 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 / µm2 ) • How many ions flow through each channel? 7-8 nm V~0.1 V V~-0.02 V Plasma membrane Cytoplasm Na+ Cl- 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 -> 2.6x107 ions flow (100 channels/µm2)x4π(50 µm)2=3.14x106 ion channels Ion flow / channel =(2.6x107 ions) / 3.14x106 channels ~ 7 ions/channel Physics 208 Lecture 10

- - - - - - + + + + + + Cell membrane as dielectric A- K+ Extracellular fluid • Membrane is not really empty • It has molecules inside that respond to electric field. • The molecules in the membrane can be polarized 7-8 nm Plasma membrane Cytoplasm Na+ Cl- Dielectric: insulating materials can respond to an electric field by generating an opposing field. Physics 208 Lecture 10

- - + + 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 Physics 208 Lecture 10

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 = E0/ k andV = V0/ k • The capacitance increases C = k C0 Eind E0 Physics 208 Lecture 10

- - - - - - + + + + + + Cell membrane as dielectric A- K+ Extracellular fluid • 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? 7-8 nm Plasma membrane Cytoplasm 70 7 0.7 Na+ Cl- C increases by factor of 10 10 times as much charged needed to reach potential Physics 208 Lecture 10

Last time… Fields, forces, work, and potential