Last time

1
Last time Fields, forces, work, and potential

• Electric forces and work

Electric potential energy and electric potential
2
Exam 1

• Average 79
• Letter grades indicate how you should interpret
  this percentage
• Average is at the B/BC border.

3
Last time Fields, forces, work, and potential

• Electric forces and work

Electric potential energy and electric potential
4
Electric field from potential

• Said before that

• Spell out the vectors

• This works for

Usually written
5
Quick Quiz

• Suppose the electric potential is constant
  everywhere. What is the electric field?

1. Positive
2. Negative
3. Increasing
4. Decreasing
5. Zero

6
Potential from electric field

• Electric field can be used to find changes in
  potential
• Potential changes largest in direction of
  E-field.
• Smallest (zero) perpendicular to E-field

VVo
7
Equipotential lines

• Lines of constant potential
• In 3D, surfaces of constant potential

8
Quick Quiz
How does the electric potential outside a uniform
infinite sheet of positive charge vary with
distance from the sheet?

1. Is constant
2. Increasing as (distance)1
3. Decreasing as (distance)1
4. Increasing as (distance)2
5. Decreasing as (distance)2

9
Electric Potential - Uniform Field
Constant E-field corresponds to linearly
decreasing (in direction of E) potential Particle
gains kinetic energy equal to the potential
energy lost
10
Check of simple cases

• Previous quick quiz uniform potential
  corresponds to zero electric field
• Constant electric field corresponds to linear
  potential

11
Complicated check point charge

• E points opposite to direction of steepest slope
• Magnitude proportional to local slope

12
Potential of spherical conductor

• Charge resides on surface, so this is like the
  spherical charge shell.
• Found E keQ / R2 in the radial direction.
• What is the electric potential of the conductor?

13
Quick quiz
So conducting sphere of radius R carrying charge
Q is at a potential

• Two conducting spheres of diff radii connected by
  long conducting wire. What is approximately true
  of Q1, Q2?

14
Connected spheres

• Since both must be at the same potential,

Charge proportional to radius
Surface charge densities?
Surface charge density proportional to 1/R
Electric field? Since ,
Local E-field proportional to 1/R (1/radius of
curvature)
15
Varying E-fields on conductor

• Expect larger electric fields near the small end.
  Electric field approximately proportional to
  1/(local radius of curvature).
• Large electric fields at sharp points, just like
  square (done numerically previously)
• Fields can be so strong that air ionized and ions
  accelerated.

16
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




17
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

1. is Vo
2. is 2Vo
3. is 4Vo
4. depends on shape

18
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.

19
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

20
Definition of Capacitance

• Same as for single conductor
• but ?V potential difference between plates
• Q charge transferred between plates
• The SI unit of capacitance is the farad (F)
• 1 Farad 1 Coulomb / Volt
• This is a very large unit typically use
• mF 10-6 F, nF 10-9 F, pF 10-12 F

21
Parallel plate capacitor
Q
-Q
outer
inner

• Charge Q moved from right conductor to left
  conductor
• Each plate has size Length x Width Area A
• Plate surfaces behave as sheets of charge, each
  producing E-field

d
22
How did the charge get transferred?

• 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

DV
23
Parallel plate capacitor
-?

• Charge only on inner surfaces of plates.
• E-field inside superposition of E-field from each
  plate.
• Constant E-field inside capacitor.

?
d
24
What is the potential difference?

• Electric field between plates
• Uniform electric field

Etotal
Potential difference V-V- (1/q)x(- work to
move charge from to minus plate)
d
-Q
Q
25
What is the capacitance?
-Q
Q
This is a geometrical factor
d
26
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
27
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
28
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 / µm2 )
• How many ions flow through each channel?

K
A-
Extracellular fluid
7-8 nm
?V0.1 V
?V-0.02 V
Plasma membrane
Cytoplasm
Na
Cl-
Charge xfer required ?QC?V(35 pF)(0.12V) (35x10
-12 C/V)(0.12V) 4.2x10-12 Coulombs 1.6x10-19
C/ion -gt 2.6x107 ions flow
(100 channels/µm2)x4p(50 µm)23.14x106 ion
channels
Ion flow / channel (2.6x107 ions) / 3.14x106
channels 7 ions/channel
29
Cell membrane as dielectric
K
A-
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.
30
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

E0
E
31
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 and V V0/ k
• The capacitance increases C k C0

Eind
E0
32
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?

K
A-
Extracellular fluid
7-8 nm
Plasma membrane
Cytoplasm

1. 70
2. 7
3. 0.7

Na
Cl-
C increases by factor of 10 10 times as much
charged needed to reach potential
33
Charge distributions
-Q

• -Q arranged on inner/outer surfaces of outer
  sphere.
• Charge enclosed by Guassian surface -QQ0
• Flux through Gaussian surface0, -gt E-field0
  outside
• Another Gaussian surface
• E-field zero inside outer cond.
• E-field zero outside outer cond.
• No flux -gt no charge on outer surface!

Q
34
Spherical capacitor
Charge Q moved from outer to inner sphere Gauss
law says EkQ/r2 until second sphere Potential
difference
Along path shown
35
Work done and energy stored

• During the charging of a capacitor, when a charge
  q is on the plates, the work needed to transfer
  further dq from one plate to the other is
• The total work required to charge the capacitor
  is
• The energy stored in any capacitor is
• 
  For a parallel capacitor

U 1/2 ?oAdE2
36
Energy density

• The energy stored per unit volume is
• U/(Ad) 1/2 ?oAdE2
• This is a fundamental relationship for the energy
  stored in an electric field valid for any
  geometry and not restricted to capacitors

37
Quick Quiz 1
A parallel plate capacitor given a charge q. The
plates are then pulled a small distance further
apart. Which of the following apply to the
situation after the plates have been moved?
1)The charge decreases 2)The capacitance
increases 3)The electric field increases 4)The
voltage between the plates increases 5)The
energy stored in the capacitor increases
C e0A/d ? C decreases!
E Q/(e0A) ? E constant
V Ed ? V increases
U QV / 2 Q constant, V increased? U
increases
38
Capacitors in Parallel

• Both ends connected together by wire

• Add Areas Ceq C1C2
• Share Charge Qeq Q1Q2

Veq

• Same voltage V1 V2

15 V
15 V
15 V
C1
C2
10 V
10 V
10 V
39
Capacitors in Series

• Same Charge Q1 Q2 Qeq
• Share VoltageV1V2Veq
• Add d

Q


C1


-




C2
-Q
-

-



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Electric Dipole alignment

• The electric dipole moment (p) along line
• joining the charges from q to q
• Magnitude p aq
• The dipole makes an angle ? with a uniform
  external field E
• The forces FqE produce a net torque t p x E
• of magnitude t Fa sin ? pE sin ?
• The potential energy is work done by the torque
  to rotate dipole dW t dq
• U - pE cos ? - p E
• When the dipole is aligned to
• the field it is minimum
• U -pE equilibrium!

42
Polar Molecules

• Molecules are said to be polarized when a
  separation exists between the average position of
  the negative charges and the average position of
  the positive charges
• Polar molecules are those in which this condition
  is always present (e.g. water)

43
How to build Capacitors

• Roll metallic foil interlaced with thin sheets of
  paper or Mylar

• Interwoven metallic plates are immersed in
  silicon oil

Electrolitic capacitors electrolyte is a
solution that conducts electricity by virtue of
motion of ions contained in the solution
44
Capacitance of Parallel Plate Capacitor

• The electric field from a charged plane of charge
  per unit area
• s Q/A is E s/2e0
• For 2 planes of opposite charge
• E s/e0 Q/(e0A)

DV
E

-
A
A
-
d
E
E
E
E-
E-
e01/(4pke)8.85x10-12 C2/Nm2
E-
45
Spherical capacitor
Capacitance of Spherical Capacitor
Capacitance of Cylindrical Capacitor
46
Charge, Field, Potential Difference

• Capacitors are devices to store electric charge
  and energy
• They are used in radio receivers, filters in
  power supplies, electronic flashes

Charge Q on plates
Charge 2Q on plates
VA VB 2E0 d
V VA VB E0 d

Potential difference is proportional to charge
Double Q ? Double V E?Q, V?E, Q?V
47
Human capacitors cell membranes

• Membranes contain lipids and proteins
• Lipid bilayers of cell membranes can be modeled
  as a conductor with plates made of polar lipid
  heads separated by a dielectric layer of
  hydrocarbon tails
• Due to the ion distribution between the inside
  and outside of living cells there is a potential
  difference called resting potential
• http//www.cytochemistry.net/Cell-biology/membrane
  .htm

48
Human capacitors cell membranes

• The inside of cells is always negative with
  respect to the outside and the DV 100 V and 0.1
  V
• Cells (eg. nerve and muscle cells) respond to
  electrical stimuli with a transient change in the
  membrane potential (depolarization of the
  membrane) followed by a restoration of the
  resting potential.
• Remember EKG!
• The Nobel Prize in Chemistry (2003) for
  fundamental discoveries on how water and ions
  move through cell membranes.
• - Peter Agre discovered and characterized the
  water channel protein
• - Roderick MacKinnon has elucidated the
  structural and mechanistic basis for ion channel
  function.
• http//nobelprize.org/nobel_prizes/chemistry/laure
  ates/2003/chemadv03.pdfsearch22membrane20chann
  els22

49
Ion channels

• Membrane channels are protein/sugar/fatty
  complexes that act as pores designed to transport
  ions across a biological membrane
• In neurons and muscle cells they control the
  generation of electrical signals
• They exist in a open or closed state when ions
  can pass through the channel gate or not
• Voltage-gated channels in nerves and muscles open
  due to a stimulus detected by a sensor
• Eg in muscles there are 50-500 Na channels per
  mm2 on membrane surface that can be opened by a
  change in electric potential of membrane for 1
  ms during which about 103 Na ions flow into the
  cell through each channel from the intracellular
  medium. The gate is selective K ions are 11
  times less likely to cross than Na
• Na channel dimension and the interaction with
  negative O charges in its interior selects Na
  ions

50
How much charge flow?

• How much charge (monovalent ions) flow through
  each open channel making a membrane current?
• Data
• Resting potential 0.1 V
• Surface charge density Q0/A 0.1 mC/cm2
• surface density of channels sC 10
  channels/mm2 109 channels/cm2
• 1 mole of a monovalent ion corresponds to the
  charge
• F Faraday Constant NA e 6.02 x 1023 x 1.6
  x 10-19 105 C/mole
• NA Avogadros number number of ions in a
  mole
• Hence surface charge density s (Q0/A)/F (10-7
  C/cm2)/(105 C/mole) 1 picomoles/cm2
• Current/area I/A s/t (10-7 C/cm2)/(10-3 s)
  100 mA/cm2
• Current/channel IC (I/A)/sC (10-4
  A/cm2)/(109 channel/cm2)
  0.1 pA/channel
• (10-13 C/s/channel)/(105 C/mole) 10-18 moles of
  ions/s in a channel ?
• (10-18 moles of ions/s)/(6.02 x 1023 ions/mole)
  6 x 105 ions/s !!

51

• Honors lecture this Friday
• Superconductivity, by yours truly. 1205 pm,
  2241 ChamberlinEveryone welcome!
• HW 2 due Thursday midnite
• Lab 2 this week - bring question sheet available
  on course web site.

52
Capacitance and Dielectrics

• This lecture
• Definition of capacitance
• Capacitors
• Combinations of capacitors in circuits
• Energy stored in capacitors
• Dielectrics in capacitors and their polarization
• Cell Membranes
• From previous lecture
• Gauss Law and applications
• Electric field calculations from Potential

53
Charge distribution on conductors

• Rectangular conductor (40 electrons)
• Edges are four lines
• Charge concentrates at corners
• Equipotential lines closest together at corners
• Potential changes faster near corners.
• Electric field larger at corners.

54
Capacitors with Dielectrics

• Placing a dielectric between the plates increases
  the capacitance
• C k C0
• The dielectric reduces the potential difference
• V V0/ k

Dielectric constant (k gt 1)
Capacitance with dielectric
55
Dielectrics An Atomic View

• Molecules in a dielectric can be modeled as
  dipoles
• The molecules are randomly oriented in the
  absence of an electric field
• When an external electric field is applied, this
  produces a torque on the molecules
• The molecules partially align with the electric
  field (equilibrium)
• The degree of alignment depends on the magnitude
  of the field and on the temperature

56
The Electric Field

• is the Electric Field
• It is independent of the test charge, just like
  the electric potential
• It is a vector, with a magnitude and direction,
• When potential arises from other charges,
  Coulomb force per unit charge on a test charge
  due to interaction with the other charges.

Well see later that E-fields in electromagnetic
waves exist w/o charges!
57
Electric field and potential
Said before that

• Electric field strength/direction shows how the
  potential changes in different directions
• For example,
• Potential decreases in direction of local E field
  at rate
• Potential increases in direction opposite to
  local E-field at rate
• potential constant in direction perpendicular to
  local E-field

58
Potential from electric field

• Electric field can be used to find changes in
  potential
• Potential changes largest in direction of
  E-field.
• Smallest (zero) perpendicular to E-field

VVo
59
Quick Quiz 3

• Suppose the electric potential is constant
  everywhere. What is the electric field?

1. Positive
2. Negative
3. Zero

60
Electric Potential - Uniform Field

E cnst

• Constant E-field corresponds to linearly
  increasing electric potential
• The particle gains kinetic energy equal to the
  potential energy lost by the charge-field system

61
Electric field from potential

• Said before that
• Spell out the vectors
• This works for

Usually written
62
Equipotential lines

• Lines of constant potential
• In 3D, surfaces of constant potential

63
Electric Field and equipotential lines for and
- point charges

• The E lines are directed away from the source
  charge
• A positive test charge would be repelled away
  from the positive source charge

The E lines are directed toward the source
charge A positive test charge would be attracted
toward the negative source charge
Blue dashed lines are equipotential
64
The Electric Field

• is the Electric Field
• It is independent of the test charge, just like
  the electric potential
• It is a vector, with a magnitude and direction,
• When potential arises from other charges,
  Coulomb force per unit charge on a test charge
  due to interaction with the other charges.

Well see later that E-fields in electromagnetic
waves exist w/o charges!
65
Electric field and potential
Said before that

• Electric field strength/direction shows how the
  potential changes in different directions
• For example,
• Potential decreases in direction of local E field
  at rate
• Potential increases in direction opposite to
  local E-field at rate
• potential constant in direction perpendicular to
  local E-field

66
Electric Field and equipotential lines for and
- point charges

• The E lines are directed away from the source
  charge
• A positive test charge would be repelled away
  from the positive source charge

The E lines are directed toward the source
charge A positive test charge would be attracted
toward the negative source charge
Blue dashed lines are equipotential
67
Qinner
Qouter

• Q QinnerQouter
• Total E-field Eleft plateEright plate

In between plates,


d
-Q
Q