Last time…
Fields, forces, work, and potential
+
+
Electric forces and work
Potential energy stored
in electric field
Tues. Oct. 6, 2009
Physics 208 Lecture 10
1
Work, KE, and potential energy
When particle is not isolated,

Wexternal  K  U
Work done
on system
Change in
kinetic energy
Change in
electric potential energy
Works for constant electric field if U  qE  r



Only electric potential energy difference
 point is chosen
Sometimes a reference
E.g. U r   0 at r  (0,0,0)
 Then U r   qE  r
for uniform electric field

Tues. Oct. 6, 2009
2
Electric potential V


Electric potential difference V is the electric
potential energy / unit charge = U/q
For uniform electric field,
U r  qE  r
V r  

 E  r
q
q
This is only valid for a uniform electric field

Tues. Oct. 6, 2009
3
Check for uniform E-field
Push particle against E-field, or across E-field
Which requires work?
+
Increasing electric
potential in this direction
Tues. Oct. 6, 2009
Constant electric potential
in this direction
+
Decreasing electric
potential in this direction
4
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?
A. +2 mJ
B. -20 mJ
C. +8 mJ
D. +100 mJ
E. -100 mJ
Tues. Oct. 6, 2009
5
Potential from electric field
dV  E  d

V  Vo

d
d
E

d
V  Vo  E d

V=Vo
 V  Vo  E d

dV largest in direction of E-field.
dV smallest (zero)
 perpendicular to E-field
Tues. Oct. 6, 2009
Physics 208 Lecture 10
6
Electric potential: general
U 
F
Coulomb
 ds 
 qE  ds  q  E  ds
Electric potential energy difference U
proportional to charge q that work is done on
U /q  V  Electric potential difference
 E  ds
Depends only on charges that create E-fields

Electric field usually created 
by some charge
distribution.


V(r) is electric potential of that charge distribution
V has units of Joules / Coulomb = Volts
Tues. Oct. 6, 2009
7
Electric potential of point charge

kQ
Electric field from point charge Q is E  2 rˆ
r

What is the electric potential difference?
V 
end

r final
E  ds 
start
 k
Define V r    0

Tues. Oct. 6, 2009

Q
k
dx
 
2
rinitial r
r final
Q
Q
Q
k
k
r rinitial
rinital
rfinal
Then
Q
V r  k
for point charge
r
8
Equipotential lines


Lines of constant potential
In 3D, surfaces of constant potential
Tues. Oct. 6, 2009
Physics 208 Lecture 10
9
Topographic map

Each lines is
constant elevation
Same as constant
gravitational potential
gh (energy = mgh)

Height interval between lines constant

Tues. Oct. 6, 2009
Physics 208 Lecture 10
10
Electric field from potential

Said before that dW  Fext  ds  FCoulomb  ds 
dV  E  ds


Spell out the vectors:
 for
This works
dV  E x dx  E y dy  E z dz
dV
dV
dV
Ex  
, Ey  
, Ez  
dx
dy
dz

Usually written

Tues. Oct. 6, 2009
dV dV dV 
E  V   ,
,

 dx dy dz 
Physics 208 Lecture 10
11
Quick Quiz
Suppose the electric potential is constant
everywhere. What is the electric field?
A) Positive
B) Negative
C) Increasing
D) Decreasing
E) Zero
Tues. Oct. 6, 2009
Physics 208 Lecture 10
12
Electric Potential - Uniform Field
dV  E  ds
 VB  VA 

B
B
B
 E  ds   Exˆ  ds
A
   Edx  E
A
B
A
 dx  E x
B
 xA 
A
B

A

Constant E-field corresponds to linearly
decreasing (in direction of E) potential
Here V depends only on x, not on y
x
Tues. Oct. 6, 2009
Physics 208 Lecture 10
13
Check of basic cases

Previous quick quiz: uniform potential
corresponds to zero electric field
E  V  constant   0

Linear potential corresponds to constant
electric field
 



E  V  Ex    Ex, Ex, Ex  Exˆ
y
z 
x
Tues. Oct. 6, 2009
Physics 208 Lecture 10
14
Potential and charge


Have shown that a conductor has an electric
potential, and that potential depends on its charge
For a charged conducting sphere:
+ + +
+
+
+
+
+
+
+ +
Tues. Oct. 6, 2009
Q k
V R V   k  Q
R R
Electric potential proportional
to total charge
Physics 208 Lecture 10
15
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
Tues. Oct. 6, 2009
Physics 208 Lecture 10
16
Capacitance

Electric potential of any conducting object
proportional to its total charge.
1
V Q
C
C = capacitance
  Large capacitance: need lots of charge to change potential


Small capacitance: small charge can change potential.
Tues. Oct. 6, 2009
Physics 208 Lecture 10
17
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
Tues. Oct. 6, 2009
Physics 208 Lecture 10
18
Definition of Capacitance

Same as for single conductor
1
V  Q
C


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
Tues. Oct. 6, 2009
Physics 208 Lecture 10
19
How was charge transferred?


Battery has fixed electric potential difference
across its terminals
Conducting plates connected to battery
terminals by conducting wires.

Vplates = Vbattery across plates

Electrons move

V

from negative battery terminal to -Q plate

from +Q plate to positive battery terminal
This charge motion requires work

The battery supplies the work
Q  CV
Tues. Oct. 6, 2009
Physics 208 Lecture 10
20
Work done to charge a capacitor


Requires work to transfer charge dq from one plate:
q
dW  Vdq  dq
C
Total work = sum of incremental work
Q

W

0

q
Q2
dq 
C
2C
Work done stored as potential energy in capacitor
Q2 1
1
2

U
 QV  CV 
2C 2
2
Tues. Oct. 6, 2009
Physics 208 Lecture 10
21
Example: Parallel plate capacitor
+Q



Charge Q moved from right outer
conductor to left conductor
-Q
inner
Charge only on inner surfaces
Plate surfaces are charge sheets,
each producing E-field
E left  E right   /2o   /2o   /o
Uniform field between plates
d
Tues. Oct. 6, 2009
Physics 208 Lecture 10
22
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
Tues. Oct. 6, 2009
Physics 208 Lecture 10
23
What is potential difference?
Potential difference = V+-V= - (work to move charge q
from + plate to plate) / q
 qEd/q
 d 
V  Ed  d /o  Q 
o A 

 d 
V  V  V  Q 
o A 
Tues. Oct. 6, 2009
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Q
Physics 208 Lecture 10
E
d
-
-Q
24
What is the capacitance?
 d 
V  V  V  Q 
o A 
V  Q /C
C
o A
d
-Q
This is a geometrical factor
+Q
Energy stored in parallel-plate capacitor
1
1 o A
1
2
2
U  CV  
Ed  Ado E 2
2
2 d
Energy density
Tues. Oct. 6, 2009
2
1
U /Ad  o E 2
2
Physics 208 Lecture 10
d
25
Human capacitors

Cell membrane:

100 µm


‘Empty space’ separating
charged fluids (conductors)
~ 7 - 8 nm thick
In combination w/fluids, acts as
parallel-plate capacitor
Extracellular fluid
Plasma membrane
Cytoplasm
Tues. Oct. 6, 2009
Physics 208 Lecture 10
26
Modeling a cell membrane
A-
K+

Extracellular fluid
+ + + + + +
7-8
nm

V~0.1 V
Plasma membrane
- - - - - -

Cytoplasm
Na+

Cl-
Ionic charge at surfaces of
conducting fluids
Capacitance:
o A
d
Tues. Oct. 6, 2009
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
100 µm sphere ~ 3x10-4 cm2
surface area
8.85 10


12
F /m4  50 10 m
6
8 109 m
Physics 208 Lecture 10
2
 3.5 1011 F  35 pF
~0.1µF/cm2
27
Cell membrane depolarization
K+
A-

Cell membrane can reverse potential
by opening ion channels.

Potential change ~ 0.12 V
Extracellular fluid
- +
- +
- +
- +
- +
+
7-8
nm

Ions flow through ion channels
V~0.1
V
V~-0.02
V
Plasma membrane
Channel spacing ~ 10xmembrane
thickness (~ 100 channels / µm2 )
+
- +
- +
- +
- +
- +
Cytoplasm

How many ions flow through each
channel?
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)x4(50 µm)2=3.14x106 ion channels
Ion flow / channel =(2.6x107 ions) / 3.14x106 channels ~ 7 ions/channel
Tues. Oct. 6, 2009
Physics 208 Lecture 10
28
Cell membrane as dielectric
K+
A-
Extracellular fluid
+ + + + + +
7-8
nm
Plasma membrane

Membrane is not really
empty

It has molecules inside that
respond to electric field.

The molecules in the
membrane can be polarized
- - - - - Cytoplasm
Na+
Cl-
Dielectric: insulating materials can respond to an electric
field by generating an opposing field.
Tues. Oct. 6, 2009
Physics 208 Lecture 10
29
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
Tues. Oct. 6, 2009
+
E
Physics 208 Lecture 10
30
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
Tues. Oct. 6, 2009
Physics 208 Lecture 10
Eind
E0
31
Cell membrane as dielectric
K+
A-
Extracellular fluid
+ + + + + +
7-8
nm
Plasma membrane
- - - - - -

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
Na+
A. 70
Cl-
B. 7
C. 0.7
C increases by factor of 10
10 times as much charged needed to reach potential
Tues. Oct. 6, 2009
Physics 208 Lecture 10
32