2b electricity - ucsd

8/9/2019 2B Electricity - UCSD

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5/16/2013 1

Physics 2B:

Electricity and

Magnetism

Eric L. Michelsen

emichels at physics etc.

Part I: Electricity

Slightly tailored for Randall Knight’s

Physics for Scientists and Engineers, 2nd

ed.



5/16/2013 2

Charges

• Do particles push on each other because theyare charged? Or ...

• Do we say that particles are charged because

they push on each other?

+ - particle 1 particle 2

F21 F12

+ +1 2

F21 F12



5/16/2013 3

More on charges• Charge is measured in Coulombs (C)

• A coulomb is like a “dozen,” only bigger:6.242e18 electrons-worth of charge

• ~10-5 moles of electrons

• Or, an electron has 1.602e-19 C of charge• Very small

+ -1 2

F21 F12

+2 -

1 2

F12

?

+2 -3

1 2

F12

?

Force is proportional

to both charges:

|F| = k e q1q2 / r 2

fromparticle 2 on particle 1



5/16/2013 4

Coulomb’s Law

• Force weakens with distance• α 1/r 2 (like gravity)

• Obeys superposition

• New charges don’t interfere with forces fromexisting charges, so forces simply add vectorially

+ -1 2F21 F12

+ -F21 F12

?

1 212 122

1212 12

0

ˆ

ˆ,

1

4

e

e

q qk

r

where r r

k K

F r

rr rr12

r12



5/16/2013 5

What are the units of ?

A dimensionless

B m

C m2

D m-1

E m-2

1212ˆ

r

rr



5/16/2013 6

Aside:

Superposition

• Kinematic example• First: Take conditions one

at a time

• Only initial position

• Only initial velocity• Only constant acceleration

• Superposition:

• The effect of all the

conditions together is thesum of each separately

• Often used in E&M

x(t )

t

v(t )

t

v(t )

t

t

xi x f = xi

x f = area = vit

0

t 0

t 0

v f =

a t

x f = area = (1/2)at 2

vi = 0, a = 0

xi = 0, a = 0

xi = 0, vi = 0

x f = xi + vit + (1/2)at 2


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Source charges and test charge

• Source charges usually considered fixed in space• Test charge moves around

• No physical difference between “source” and “test”

• Coulomb’s Law obeys superposition

+

-

-

+

test chargeq0 > 0

12

3

4 F1

F2

F3

F4

Fesource

charges

1-4

40

21

4

0 21

ˆ

ˆ

i

e e iii

ie i

ii

q q

k r

qq k

r

F r

r



Electric Fields

• Consider a distribution of source charges• The force on a test charge at any point is

proportional to the charge:

0 0

0

At a point: constant vector

Extending to any point in space: ( ) ( )

e e

e

q q where

q

F F E E

F r E r

+

--

+

q0 < 0

12

3

4 F1

F2

F3

F4

Fe

+

--

+

q0 > 0

12

3

4 F1

F2

F3

F4

FeUnits

of E?

Direction

of E?

There’s no

physical

difference

between source

and test charges



Electric field from source charges

• Source charges create an E-field4

0 021

4

21

ˆRecall: ,

ˆ( )

ie e i e

ii

ie i

ii

qq k and q

r

qk r

F r F E

E r r

+

-

-

+

q0 > 0

12

3

4 F1

F2

F3

F4

Fe



At P, which way does

the E-field point?

A

B

C

D

E

-

-

P

●



Lines of Force

• Field of vectors in space: the E-field• Lines of force follow the arrows

E

How does lines-of-

force density relate

to E-field strength?

E-field strength is

proportional to lines-

of-force density.

+ -



Do objects always move in the

direction of the force applied to them?

• No. Objects bend toward the direction ofapplied force, but they generally don’t move

parallel to it

• Lines of force• Not trajectories of motion

• g-field?

F g



5/16/2013 13

Simulated particle trajectory

• What sign is the test charge?

line offorce

trajectory



5/16/2013 14

If I release a negative test charge at rest

on a line of force, how will it move?

A To the right, following the line of force

B To the right, crossing the lines of force

C It won’t moveD To the left, following the line of force

E To the left, crossing the lines of force

+ -

When does the

test charge move

along the line of

force?



5/16/2013 15

Birth, life, and death of

lines of force

• No matter how close we get to a positive charge, lines of force point away from it• Lines of force originate on positive charges

• No matter how close we get to a negative charge, lines of force point toward it

• Lines of force terminate on negative charges

• Can’t cross or split• ... because they follow the E-field, which can only point one way from

every point

• But they can turn quickly at a point of zero field

• •

+ −

X • •E = 0



5/16/2013 16

Electric conductors

• Conductors (good or poor) have mobile charges• Good conductors: metal

• Poor conductors: carbon, electrolytes (e.g. water)

• If immersed in an electric field, mobile charges move

• While they are moving, we have an electric current

• current is the motion of charge

q

-q I

velocity

Much more

on this later



5/16/2013 17

Insulators

• Insulators do not have long-range mobile charges

• E.g., glass, plastic

• May have dipoles that can be

induced or rotate a little

• All insulators are dielectric at

some level

• Some insulators have negligible

polarization (gases)

• Don’t worry about dielectrics

until a later chapter Much more

on this later

appliedE



5/16/2013 18

Charging by induction

• Start withneutralconductors

• Induce• Connect

spheres by wireor touching

• Separate• (E-field

not shown)

+-+

+-+

Connect

with wire

A l t h + h it



5/16/2013 19

An electroscope has a + charge, so its

leaves are apart. I bring a negative rod

near it. What happens to the leaves?A Leaves get closer

B Leaves spread farther

C One gets higher, the other lower

D Nothing

E Leaves collapse completely

+

- -

-

The electroscope question

(stop to think 26.3 p799)How can we be

confident this is true?



5/16/2013 20

dQ

E-field of ring of charge

• What is E-field at distance z from center?• Axial symmetry implies E parallel to axis

R

z

3 / 22 2

ˆ,e z

z k Q E

R z

E z

Q

r

d E

dE z

θ θ

2 2ˆ cose z e

dq dqd k dE k

r r E r

2 2 2 2 2

z e e

dq z z dE k k dq

r r R z R z

3 / 20 02 2

z E Q

z e

z dE k dq

R z



5/16/2013 21

Flux• Loosely: “counting” lines of force through the area

• This only works because E α 1/r 2

• More precisely: Flux is E-field component through the area,times the area:

• Area vector is perpendicular to the area, so the parallel E-field

component gives the E-field perpendicular to the area• So flux is a dot-product

E

A(area vector)

E E area

or d E A E A

E

AWhat kind of quantity

is Φ ?



5/16/2013 22

Why flux?• Surface charge distributions (sheet

charge)• A uniform density of charge in a large

flat sheet

• What is the E-field near the surface?• Critically important, but ...

• Can integrate over Coulomb’s Law, but it’stedious

• Gauss’ Law (flux) makes this simple

+ + + +

+ + + +

+ + + +

sheet charge:η C/m2

side

view

E

+ + + +

F th |E| d |A| h d th



5/16/2013 23

For the same |E| and |A|, how does the

flux through the area on the left

compare to that on the right?A Flux is greater on the left

B They are the same

C Flux is greater on right

E

A

E

A



5/16/2013 24

What is the flux through a sphere?

A

B

C

D

E

q

E E A

2

e E

k q

r

2ˆe

E

k q

r

r

r

24 E er k q

4 E ek q

0 E d Ω?

d A E

surface

cosd d E dA E A

2

2

surface

4 4e e

qdA k r k q

r



5/16/2013 25

What is the flux through the funny-

shaped surface?

q

A

B

C

D

E

4 E ek q

0 E

4 E ek q

4 E ek q

r



5/16/2013 26

What is the flux through the bag?

A

B

C

D

Eq

4 E ek q

0 E

4 E ek q

4 E ek q



5/16/2013 27

Gauss’ Law

• The total outward flux from a closed surfaceequals 4πk e times all the charge enclosed:

• Does superposition apply?

• Superposition of what?

• What is qin here?

0

0 0

14 , will be handy later

4

in E e in e

qk q k

q

2q

Why are arrows

different lengths?



5/16/2013 28

Gauss’ Law (part deux)

q

out +

in -

out +

out +

S f h di ib i



5/16/2013 29

Surface charge distributions:

the beauty of Gauss’ Law

• A uniform density of charge in a large flatsheet

• Gazillions of tiny particles (electrons, ions)

• What is the E-field near the surface?• What “shape” is it?

+ + + +

+ + + +

+ + + ++ + + +

sheet charge

density, η C/m2

gaussian

surface (box)

side

view

E

0

00

Gauss' Law: /

2 / ,2

E inq

EA A E

A

A

A



5/16/2013 30

What is the E-field above a surface-

charged conductor?

A zero

B η/2ε0

C η/ε0

D 2η/ε0

E 4η/ε0

conductor

+ + + + + + + + +

E?

How is this possible?η



5/16/2013 31

Three ways to make E-fields easy

• Using Gauss’ Law relies on any of 3 methods• E-field is constant (though unknown)

• E-field is zero

• E-field is tangent (to gaussian surface)

+ + + + + + + + +

E? A

AE = 0

conductor



5/16/2013 32

Volume charge distributions

• Consider a spherically symmetric volumedistribution, in C/m3

• What is E-field at distance r , outside thesphere?

• What does Gauss say?

• How does it compare toa point charge at distance r ?

q

r

2

2

4 4 4 E e in r earea

r e

k q E r k q

q E k

r

H d h E fi ld f h i ll



5/16/2013 33

How does the E-field of a spherically

symmetric volume distribution

compare to that of a point charge?A It’s bigger, because some of the charge is

closer

B It’s the same

C It’s smaller, because more of the charge is

farther away

qr r

F l d t k t ti l



5/16/2013 34

Forces lead to work, potential energy,

and voltage (aka potential)

• 1D:• Independent of the trajectory: time, speed

• Work can be + or -

• Potential energy is proportional to test charge

• Voltage is defined as electric potential energy perunit charge (aka potential difference)• In volts (V), equivalent to joules/coulomb

0( ) ( ) ( ) ( )

B B

e A AW F x dx U B U A W q E x dx

q0 E > 0

F

A B

0

( ) ( )( ) ( ) ( 0)

B

A

U B U AV B V A E x dx

q

x

q0 E > 0

F

A B

+

-

“El t i P t ti l” i diff t th



5/16/2013 35

“Electric Potential” is different than

“Potential Energy”

• Electric Potential (aka “Potential”) = Voltage• Potential is defined as electric potential energy per

unit charge

• Potential is independent of test charge

• “Potential energy” of the test charge depends

on its charge:

0

( ) ( )( ) ( ) e eU B U A

V B V Aq

0( ) ( ) ( ) ( )e eU B U A q V B V A

H d th k d (f A t B)



5/16/2013 36

How does the work done (from A to B)

by the field on a fast-moving charge

compare to the work done on a slow-moving charge?

A It’s greater

B It’s the same

C It’s less

D Depends on the charge

E > 0 A B

q0



5/16/2013 37

Potential of a point charge

• In other words, what is the work needed to bring +1 C ofcharge from far away to a distance r from the source charge?• Gets steeper closer in

• Which way does a positive charge move?

+ r

V (r )

-

V (r )

r E(r )

PE of +

test charge

PE of -

test charge V (r ) < 0



5/16/2013 38

Potential of a point charge (2)

• In other words, what is the work needed to bring +1 C ofcharge from far away to a distance r from the source charge?

• We can use a 1D analysis

• Voltage referenced to ∞ is defined as absolute potential

• Still a function of distance, r :

• If source charge is negative, potential is negative

qF A at ∞

Bq0displacement

direction from A to B

2

1( ) ( ) ( )

B B

r B r

e e e A B

q qV B V A E r dr k dr k q k

r r r

( ) B B r e

B

qV r V k

r

source

charge

V (r )



5/16/2013 39

What is the potential midway

between two point charges?

A positive

B zero

C negativeD depends on the test charge

q -qq0

V ?

Wouldn’t I have to do work to bring a



5/16/2013 40

Wouldn t I have to do work to bring a

charge from infinity to the midpoint?

• At first, from some directions, yes• But once past the + charge, the E-field does as muchwork (returns as much energy) to finish the job as itcost to bring the charge in from infinity• Net energy cost = 0

q -q

maximum potential

equipotentialsurfaces



How does the work done on q0



5/16/2013 42

How does the work done on q0

from A to B along path 1

compare to that of path 2?A It’s more

B It’s the same

C It’s less

E

A •

• B

q0

•

p a t h 1

p a t h

2What if it were

different?

H d i h



5/16/2013 43

How do negative test charges move

in an electric potential?

A Down the potential hill

B They don’t move

C Up the potential hill

How do negative

charges move in a

potential energy field?

V ( x)

?

U (r)?

Different setup

than above

x

When acted on only by the potential field.



5/16/2013 44

E and V

• E is to F as V is to U e• Electric field E is force per unit charge

• V is potential energy per unit charge

0 0

( ),( ) )( () e eU q q

V rF rE r r electric potential energy per unit charge

electric force per unit charge



5/16/2013 45

Hard work: grading your potential

x

y

z

dx

dy

dz

,e x xđW F dx

,e z z đW F dz

,e y yđW F dy

any path

( ) ( ) ( ) B

A

dV d

V B V A d

E s

E r s

ˆ ˆ ˆd dx dy dz s x y z ( ) ( ) ( )ˆ ˆ ˆ( )

( )

U U U

x y z

U

r r rF r x y z

r

The gradient produces a vector

field from a scalar field

0 0

( ) ( )

( ) ( )

U

q q

F r r

E r V r

d s

E

A •

• B

d s

Equivalence of statements



5/16/2013 46

Equivalence of statements

about conservative fields• “The E-field is conservative”

• ≡ “The electric potential is a function of position only: V (r )”

• ≡ “The electric work done on a charge between two points isindependent of path”

• We expect this from conservation of energy

• Different work on different paths + reversing a path reverses the work= infinite energy for free!

• Can prove it explicitly from Coulomb’s Law & vector calculus• Specifically: the curl of the electrostatic E-field = 0

0 E

0

0

d

B s

B



5/16/2013 47

Good conductors

• Good conductors (typically metals) have huge(essentially infinite) numbers of mobile electrons(~104 C/cm3)

• Electrons free to move within the conductor

• But they can’t leave the surface

• E-field drives negative electrons• leave behind positive ions

• Allows for a redistribution of charge

• Charge always moves to reduce E-field

• Molarity:

• 1 C = 10-5 mol of electrons

• Copper mobile electron density= 10-1 mol/cm3 = 100 mol/L

applied E

+

+

+

-

-

-

I d t hi h d th



5/16/2013 48

In a conductor, which way do the

positive charges move?

A left

B they don’t move

C rightD depends on the conductor

applied E

+

+

+

-

-

-

In a static E-field, after equilibrium,



5/16/2013 49

In a static E field, after equilibrium,what is the E-field inside a good

conductor?A Depends on the source charges around it

B Enhances any applied E-field

C Reduces any applied E-fieldD zero

applied E+

+

+

-

-

-E?

conductor

I ilib i h d h i l



5/16/2013 50

In equilibrium, how does the potential

at B, V (B), compare to that at A, V (A)?

A It’s less

B It’s the same

C It’s greater D Depends on the conductor

applied E+

+

+

-

-

-A •

• B

conductor



5/16/2013 51

Fluid conductors

• Have smaller number of mobile charges• Mobile charges may be electrons or ions

• Conductor may be liquid or gas

• Plasma (ion/electron gas, not blood)• Electrolyte

Whi h d th iti h



5/16/2013 52

Which way do the positive charges

move?

A left

B they don’t move

C right

applied E



5/16/2013 53

Alternate units for E

• [E]=V/m• Equivalent to N/C

• Can view E two ways:

• For simplicity, consider

a uniform E-field:

0 0e V U q q FE d d

force

voltage

, or fromV V d E d E x

E

d

+ ΔV − A B

potential energy

per unit charge

Sign convention for ΔV :

positive end of ΔV has

higher PE

for positive test charge

?

sign convention

Relationship between E-field and V-field



5/16/2013 54

Relationship between E-field and V-field

• E points between equipotentials

• E is perpendicular to equipotentials• Tighter equipotential surfaces = stronger E (more volts/meter)

• Wider equipotential surfaces = weaker E (less volts/meter)

q -q

equipotentialsurfaces

E

)( () V E r r

1 V

2 V 3 V

)( () U F r r

0 V

−3 V

decreasing

potential: E points “downhill”

E



5/16/2013 55

Potential energy of a bunch of charges

• Imagine bringing charges, q j, in one at a time• The 2nd and subsequent charges ( j)interact with all previous charges (i)

• Also written (as in Knight):

• But it’s really a double sum

• And even:

1

2 1

,

jni j

e e ij ijij j i

q qU k r

r

r

i je e

iji j

q qU k

r

q1 q2

q3 q4

1

2

i je e

iji j

q qU k

r



5/16/2013 56

Capacitor

• Two parallel thin plates• Small gap =>

neglect edges

• Charges symmetrically• By induction!

• (not to be confused withinductance)

• Stores charge

• Q (charge on one plate)is proportional to ΔV

• Defines capacitance C :• Q = C ΔV

• Measured in C/V ≡ F,Farads

E+

+

+

+

−

−

−

−

wire

large plate

smallgap, d

+ ΔV −

What is the E field outside the



5/16/2013 57

What is the E-field outside the

plates of a capacitor?

A left

B zero

C right

D depends on which plate has more charge

E not enough information +

+

+

+

-

-

-

-

wire

+ ΔV −

i i



5/16/2013 58

Energy in a capacitor

• Given a capacitor charged to voltage ΔV , howmuch work does it take to increase the charge by a small amount, dq?

• How much work tocharge the wholething from 0 to ΔV ?

+

+

+

+

-

-

-

-

wire

outside

qdU dW V dq dq

C

E

hypothetical

dq > 0

NEVER-

EDDY

2

0 00

22

1

2

1 1

2 2

QW Q

outside

q qU dW dq

C C

QC V

C

+ ΔV −

final charge

+−

−dW e

NB: dq is not a

geometric quantity

Si i f i



5/16/2013 59

Sign convention for a capacitor

• Choose reference plate: defines polarities• Eliminates minus sign where possible

+

+

+

+

-

-

-

-

wire

0 0

0

0

Q E

A

d V E d Q

A

AQ V C V

d

E

d

I choose this as myreference plate

charge onreference plate

+ ΔV −

capacitance α area,

inversely α separation

What is the total net charge in a



5/16/2013 60

What is the total net charge in a

capacitor charged to voltage ΔV ?

A Qnet = C ΔV

B Qnet = (1/2)C ΔV

C Qnet = 0D Qnet = C (ΔV )2

E Qnet = (1/2)C (ΔV )2

If I charge a capacitor so that



5/16/2013 61

If I charge a capacitor so that

Q < 0, what describes its energy?

A U > 0

B U = 0

C U < 0D depends on ΔV

H l bl E l 1



5/16/2013 62

How to solve any problem: Example 1

• A bound NaOH molecule is modeled as Na+ bound to O(-2) bound to H+, in a straight line in that order. The Na+ ion is1.0e-10 m from the O(-2), which is 1.0e-10 m from the H+.How much energy does it take to remove the Na+ ion from themolecule (i.e., take it to far away), in J?• What does it look like?

• What does it want?

• What do we know about what it wants?

• What is given?

• What can we find from what is given?

• Build a bridge: what principles/formulas relate what is given or whatwe can find to what it wants?

+ -2

Na O H+

1e-10 m 1e-10 m

U q V

( ) e

qV k

r r

1 2

0 0 10 10

( ) ( ) ( )

20

1 10 2 10

total

p ptotal e

V V V

q qU q V q k

r r r

V total (∞)

ifneeded

Energy

q1, q2, q3, r 1, r 2

H l bl E l 2



5/16/2013 63

How to solve any problem: Example 2

• An electron in a picture tube starts at rest. It is accelerated by

25,000 V across a distance of 0.50 m to the screen. How longdoes it take to travel to the screen, in s?• What does it look like?

• What does it want?

• What do we know about what it wants?

• What is given?• What can we find from what is given?

• Build a bridge: what principles/formulas relate what is given or whatwe can find to what it wants?

2, ,

F qE x x d a t

m m a

E

screen

Δ x = 0.5 m

21,

2avg x at x v t

e e

V W F d qEd q V E

d

19

15

31

8

15

25,000 V /0.5 m 50,000 V/m

1.6 10 C 50,000 V/m8.8 10 m/s

9.1 10 kg

2(0.5 m)

1.1 10 s8.8 10

E

a

t

time

d , ΔV

ifneeded

Fe

When mobile charges move what



5/16/2013 64

When mobile charges move, what

do they do to the applied E-field?

A they reduce it

B nothing

C they increase it

+ - + -

Di l t i (1)



5/16/2013 65

Dielectrics (1)

• Have rotating dipoles(on springs)

• Dipoles rotate to opposeapplied E-field

• For same charge on plates,moving charges reduce E-field

• Dielectrics are not conductors• Charges move a little, but are

bound to a small region

0 0net net

E V E V

applied

E0

Enet

+

+

+

+

−

−

−

−

wire

+ ΔV −

dielectric

little E-fields oppose E0

it’s linear!

vacuum valuevacuum value

When I insert a dielectric between the



5/16/2013 66

plates of a capacitor, what happens to

the capacitance?A It decreases

B nothing

C it increasesD depends on the dielectric E

+

+

+

+

−

−

−

−

wire

large

plate

smallgap, d

+ ΔV −

dielectric

Di l t i (2)



5/16/2013 67

Dielectrics (2)

• For same charge on plates,rotating dipoles reduce

E-field

• For same charge on plates, ΔV

decreases compared to vacuum

• Coulombs per volt increases• => capacitance increases

E+

+

+

+

−

−

−

−

wire

large

plate

smallgap, d

+ ΔV −dielectric0Q

C V d

0 0net net

V E V

dielectricconstant

Cellular



5/16/2013 68

technology

“... local dielectric properties can playcrucial roles in membrane functions.”

-- Local Dielectric Properties Around

Polar Region of Lipid Bilayer

Membranes, J. Membrane Biol.

85,225-231 (1985)

“An important physical effect of cholesterol is to increase

the membrane's internal electrical dipole potential, which is

one of the major mechanisms by which it modulates ion

permeability. The dipole potential ... arises because of the

alignment of dipolar residues ... in the .. interior of the

membrane. ... its magnitude can vary from ∼100 to >400

mV.... Recent investigations have suggested that it affects

numerous different biological membrane processes.”

-- Cholesterol Effect on the Dipole Potential of Lipid

Membranes, Biophys J. 2006 June 1; 90(11).

I’m a cell, and ions are expensive. I need to



5/16/2013 69

, p

create a transmembrane potential difference.

What should I choose to perfuse my membrane?

A Cholesterol, κ = 2

B Water, κ = 80

C Air, κ = 1D It doesn’t matter

Electricity with all my heart



5/16/2013 70

Electricity, with all my heart• Electrical model allows localizing

performance of individual muscles or

small groups from multiplemeasurements on the skin

Jose Bohorquez, An Integrated-Circuit Switched-Capacitor Model and Implementation of

the Heart , Proceedings of the First International Symposium on Applied Sciences on

Biomedical and Communication Technologies, 25-28 October 2008. DOI10.1109/ISABEL.2008.4712624

+

right atrium

body arteries and veins

tricuspid valve

+

right ventricle

pulmonic valve

+

left atrium

mitral valve

+

left ventricle

aortic valve

pulmonary arteries and veins

Charging a capacitor:



5/16/2013 71

g g p

some things just take time

• To what voltage (ΔV ) does the capacitorcharge?

• Why does it stop charging?

+

+

+

+

-

-

-

-

wire

Econventionalcurrent

NEVER-

EDDY

+ ΔV −

+−

1.5 V

electroncurrent

1.5 V

+

I

Wh t i lt ?



5/16/2013 72

What is a voltage source?• An ideal voltage source maintains a constant voltage

across its terminals under all conditions

• A battery is a voltage source. So is ...• ... a generator

• Nothing is really ideal, but it’s a useful model

• When charges move to reduce the voltageacross the terminals ...• The voltage source pumps out more charge to keep the voltage up

• Pulls in charge at low potential energy, pumps out a charge at high PE• This may or may not result in a continuous cycle of charge (current)

• An ideal voltage source can ...• Supply or absorb an arbitrarily large charge

• Supply or absorb an arbitrary large current(arbitrarily large charge in arbitrarily short time)

• Supply or absorb arbitrarily lar ge energy

+

I

charge in(low PE)

charge out(high PE)

+

+

PE

Ideal voltage source (2)



5/16/2013 73

Ideal voltage source (2)

• Usually, voltage sources supply energy to the circuit

• They have a source of energy, the emf ,

which overcomes the electric force

• The voltage source does positive work onthe charges

• The E-field does negative work on the

charges

• But sources can just as well absorb

energy from the circuit

• You can charge a battery

+

I

e- out

(high PE)

e- in

(low PE)

-

-

PE

Electron flow view

Electric current



5/16/2013 74

Electric current

• Conductors (good or poor) have mobile charges• If immersed in an electric field, mobile charges move

• While they are moving, we have an electric current

• current is the motion of charge

• Conventional current is the flow of hypothetical positive

charges

q

-q I

velocity

current

What describes the electric current



5/16/2013 75

What describes the electric current

below?

A It is to the left

B It is about zero

C It is to the rightD Depends on the conductor

q

-q

-q

Continuous current



5/16/2013 76

Continuous current

• If mobile charges are continuously removed

from one side, and resupplied on the other, we

have a continuous current

• Conventional current is the flow of

hypothetical positive charges

• In C/s ≡ amperes (A)

-q

-q

wire

hypothetical

I = dQ/dt

NEVER-

EDDY

+−

0( ) lim

avg

t

Q I

t Q dQ

I t t dt

Most references use

‘i’ and ‘q’ for time-

dependent values

+ ΔV −-q

electron

I

How does the PE of the e- going into



5/16/2013 77

the poor conductor compare to the PE

of the e- coming out?

A It’s less

B It’s equal

C It’s moreD not enough information

-q

-q

wire

NEVER-

EDDY

+−

+ ΔV −-q

electron

+

-q

poorconductor

How does the PE of the hypothetical (+)



5/16/2013 78

charge going into the poor conductor

compare to the PE of it coming out?

wire

hypotheticaldQ > 0

NEVER-

EDDY

+−

+ ΔV −

q

q

q

A It’s less

B It’s equal

C It’s moreD not enough information

From now on, we always

use conventional current

when discussing circuits.

+

I

poorconductor

A wire is a good conductor. How



5/16/2013 79

much work does it take to move a

charge through it?

A exactly zero

B essentially zero

C a lotD it’s impossible to move a charge through it

Kirchoff’s current law (junction rule)



5/16/2013 80

Kirchoff s current law (junction rule)

• Conservation of charge• Sum of currents into a node

= sum of currents out of a node

• “Node” is aka “junction”

• Sum of currents into a node = 0

• This is the book’s version

• Sum of currents out of a node = 0

• Not used much

I 1 I 2

I 3

I 1 I 2

I 3

I 1 I 2

I 3

1 2 3

in out I I

I I I

1 2 3

0

0

in I

I I I

1 2 3

0

0

out I

I I I

Kirchoff’s voltage law (loop rule)



5/16/2013 81

Kirchoff s voltage law (loop rule)

• Conservative nature of electric field

• PE of charge depends on its location,

but not how it got there

• Consider PE change of +1 C of test

charge, at each node around a loop

• If possible, choose loop

direction out from battery +

• I find it easiest to start from battery -

• Final PE must equal starting PE

• PE is proportional to charge (and to voltage)

• Final voltage must equal starting voltage

+

loopdirection

1 2 3

1 2 30 0

rises drops V V V

V V V V

+ V 1 -

+ V 2 - + V 3 -

Capacitor combinations



5/16/2013 82

Capacitor combinations

• Capacitors in parallel

• When charged, how does V 1 compare to

V 2, and to V ?

• How about charges Q1 and Q2 ?

• Capacitors in series

• When charged, how does V 1 compare to

V 2, and to V ?

• How about charges Q1 and Q2 ?

V

+

C 1

C 2

V

+

C 1C 2

1 21 2 1 2

tot tot tot Q Q QQ Q Q C C C

V V V

1 21 2 1 2

1

/ / 1/ 1/

tot tot tot

tot

Q QV V V C

V Q C Q C C C

Recall:Q

C V

Q1

Q2

+ V 1 - + V 2 -

Ohm’s “Law”



5/16/2013 83

Ohm s Law

• For a large class of conductors (at constant

temperature), the current through it is proportional to the voltage across it:

• The constant of proportionality is resistance

• Requires consistency in reference directions (sign conventions)• measured in ohms (Ω)

• Some things don’t follow Ohm’s “Law”• Light bulbs ?

wire

NEVER-

EDDY

+−

+

I

conductor

V IR

R

+ ΔV −

resistors

I

+ ΔV −

Discharging a capacitor;+ΔV



5/16/2013 84

Rearranging capacitors

dq R

+ ΔV −

+ΔV −

e

qdW V dq dq

C

+ΔV −

+ΔV −

Q

0 0

0( )

e

i i

W

eq q

qdW V q dq dq

C

21

2

ie

qW

C

• There’s no R in the final result

• There was never an R to start with

• Even a wire dissipates the same energy

• When rearranging capacitors, the

wires dissipate the missing energy

• What is the energy delivered whendischarging a capacitor?

Q/2

Q/2

The nature of resistance



5/16/2013 85

The nature of resistance

What is the effect of temperature



5/16/2013 86

p

on a given conducting element?

A Higher temp usually means higher R

B Higher temp usually means lower R

C Temp has no effect on R D There’s no general trend, it varies by material

Without using any formulas, what



5/16/2013 87

g y ,

is the total resistance?

A R/2

B R

C 2 RD Depends on R

E 10 ohms

R

R

Resistor combinations



5/16/2013 88

Resistor combinations

• Resistors in parallel

• How does V 1 compare to V 2, and to V ?

• How about currents I 1 and I 2 ?

• Resistors in series

• How does V 1 compare to V 2, and to V ?• How about currents I 1 and I 2 ?

V +

R1

V

+

1 2

1 2 1 2

1/ / 1/ 1/

tot

tot tot

I I I

V V R I V R V R R R

1 2

1 2 1 21 2

tot

tot tot

V V V

V V V IR IR R R

I I I

Recall: V IR

I 1

+ V 1 - + V 2 -

R2 I 2

I tot

R2

I 2

R1

I 1

Ohmic?

What we need is “More power”



5/16/2013 89

What we need is More power

• Two terminal devices (R, C, diodes, batteries,

etc.) supply or absorb electrical energy• At what rate?

• Resistors convert electrical energy to heat

• Sign convention for sources isopposite that of all other devices

• Capacitors sometimes absorb & sometimes supply

+

I

R

+ ΔV −

J/s J/C C/s

V I

I + ΔV −

Resistors: Positive powerabsorbs energy from circuit

Sources: Positive power

supplies energy to circuit

What is the power dissipated in a



5/16/2013 90

p p

15 Ω resistor at 120 V?

A 1800 W

B 80 W

C 960 WD 8 W

E Depends on the current

V +

I

What is the power dissipated in the



5/16/2013 91

p p

diode, in W?

A 0.10

B 0.06

C 0.04D 0.02

E not enough information

3V

100 + 2V -

+

What is the power in the diode?



5/16/2013 92

p

Solution

3V

100

+

+ V R

−

+ 2V -

From the loop rule: 3 2 0 1 volt

From Ohm's law: /10

Note the

0 0.01

polarities on

amperes

Power law on diode: 2 0.01 0.02 wa

the loop rule

tts

R R

R

V V

I V

P

I

loopdirection

Defibrillator



5/16/2013 93spoons

EKG



5/16/2013 94

EKG

• Voltages measured between pairs of leads• Sample of one voltage below right

• Different pairs measure different regions of the

heart

Home electrical

i



5/16/2013 95

service

• Drip loops keep water out

• Grounding prevents arcing

• Find your grounding rod at home

drip loops

prevent this

Old-fashioned 120 V service



5/16/2013 96

15 A

earth

ground

circuitbreakers

neutral

safetyground

hot 120 V

white

black

neutral/

ground bond

15 A

:

hotneutral

etc.

Are you beingserved?



5/16/2013 97

served?• Phase-to-neutral = 120 V

• Phase-to-phase = 240 V

15 A

phase B

phase A

earth

ground

circuit

breakers

neutral

safetyground

phases A and B areopposite polarity

neutral

phase B phase A

white

black

neutral/

ground bond

etc.

Why two phases?



5/16/2013 98

Why two phases?

120 V +

+

0.25

0.25

4 A 4 A

+

0.25

0.25 4 A

0.25

4 A

120 V120 V

240 V

receptacle- ΔVwire +

Power in a resistor



5/16/2013 99

Power in a resistor

• For all (2-terminal) devices, P = (ΔV ) I

• For resistors:

• Losses in a wire vary as the square of the current inthe wire

• Less current = less loss

• It’s easy to know the current in a wire• But hard to know the voltage across it

• High V → low I → low losses

2

2/ /

V IR P IR I I R

V I V R P V V R

What is the power lost in the wires?



5/16/2013 100


A 2 W

B 4 W

C 8 WD 16 W

E 32 W

120 V +

0.25

0.25

4 A 4 A

wires

What percentage is that ofthe supplied voltage?

What is the voltage

drop in the wires?

Tip: pick your reference

directions first:

What do the currentarrows look like?

What is I ?



5/16/2013 101

What is I 2?

A 0.0 A

B 0.50 A

C 1.0 AD 2.0 A


+

+

0.25

0.25 4 A

0.25

4 A120 V

120 V

I 2




5/16/2013 102


+

+

0.25

0.25 4 A

0.25

4 A120 V

120 V

I 2 What percentage isthat of the load?

What is the voltage

drop?

A 2 W

B 4 W

C 8 WD 16 W

E 32 W

What is earth?



5/16/2013 103

• What does the light bulb do?

• No steady-state current: bulb is dark • There can be transient current

• Charges move to decrease potentialdifferences, then current stops

• Essentially, earth is a big blob ofresistive material• Two stakes in the ground have a

resistance ~100’s to 1000’s Ω• Try it!

• This makes earth a convenientreference point for voltages

• The entire power grid is tied to earthto avoid arcing (sparks)

V +

steady state I = 0

Which path does the current take?



5/16/2013 104

Which path does the current take?

A Current takes the path of least resistance, RB Current takes the path of most resistance, 2 R

C Current divides equally between both paths

D Current divides in proportion to the resistancesE Current divides in inverse proportion to the

resistances

V +

I

R

2 R

Electrical measurements V +

R



5/16/2013 105

• Measuring voltage is easy

• Just stick the probe between the points

• Measuring current is harder

• Must insert meter into the current path

• Measuring resistance requires one end of resistor beopen (so it’s similar difficulty to measuring current)

V +

R1

R3 R2V

+ I+

R1

R3 R2

V +

R1

R3 R2

Ω open

I disagree with the book’s quantitativei i d l f d ti



5/16/2013 106

microscopic model of conduction• The (average) final speed of an accelerated electron

is:

• Which means the average speed is ½ that• Book is off by a factor of 2

• For their equations to work, we must define τ as ½ themean time between collisions

• All the books do the same thing

f

F qE v a

m m

time since collision

v(t )

vaverage

τ

v f

Current-voltage relation for a resistor



5/16/2013 107

g

• What is the shape?• What is the slope?

• Knight 2nd ed. p969b is wrong:

• Ohm’s law includes signs of V and I

V

I

I = V / R, so

slope = 1/ R

I

R

+ V −

Analyzing a reducible circuit: how arethe 600 ohm and 560 ohm resistor



5/16/2013 108

the 600 ohm and 560 ohm resistor

related?

A In series

B In parallel

C neither D both


1 2 V

600 Ω

+

800 Ω

400 Ω

5 6 0 Ω

Analyzing a reducible circuit



5/16/2013 109

y g

• A circuit is reducible if we can use simple

resistor/capacitor combinations to reduce the circuitto a single source and a single resistor and/orcapacitor.

• Usually (but not always), a circuit with only one source is

reducible

1 2 V

600 Ω

+

800 Ω

400 Ω

1 2 V +

800 Ω

240 Ω

5

6 0 Ω

5

6 0 Ω

1 2 V +

800 Ω

800 Ω

1 2 V +

800 Ω 8 0 0 Ω

1 2 V +

400 Ω

Space



5/16/2013 110

• The space station has

charge dissipators• Xenon ions carry away

built-up charge

NASA says there is a

health risk if the ISS

charges to more than 40

volts, and a 2002 NASAreport says that voltages

as low as 60 volts could

cause cardiac arrest in a

spacewalking astronaut.

Sparks



5/16/2013 111

p

• Dielectric strength of air ~3e6

V/m• At ~100 V => 3e-5 m ~ 0.03 mm

electrical switch

“Drawing” current



5/16/2013 112

• Nothing “draws” current

• Devices allow current to flow if pushed by a

voltage (potential difference)

g

I



Analyzing a 2-loop circuit: which



5/16/2013 114

resistors can we combine?

A 300 and 100B 300 and 200

C 100 and 200

D All 3 at onceE None of them

19 V

300 Ω

+200 Ω

12 V +

100 Ω

I 1 I 2

Analyzing an irreducible 2-loop circuit



5/16/2013 115

• Can we reduce it?

• How many unknowns are there?• What about the unlabeled current?

19 V

300 Ω

+200 Ω

12 V +

100 Ω

I 1 I 2

? I 1 - I 2

1 1 219 300 100 12 0 I I I

1 2 212 100 200 0 I I I

1 1 2 1 2

1 2 2 1 2

1 2

1 1

2 2

7 300 100 400 100

12 100 200 100 300

21 1200 300

33 1100 0.03 A

12 3 300 0.05 A

I I I I I

I I I I

I I

I I

I I

1 2

1 2

1

Or:

19 300 200 0

19 300 200

33 1100

I I

I I

I

Current-voltage relation for acapacitor: i(t) ? v(t)



5/16/2013 116

capacitor: i(t ) ? v(t )

• Note reference polarities

• What is relation between q and i?

• What is relation between q and v?

• Finally, what is relation between i and v?

i

C

+ v −

q(t )

( )( )

dq t i t

dt

( )( ) ( ) ( )

dv t q t Cv t i t C

dt

What is the current?



What is the current?

A 64(-sin ωt ) mA

B 10.(-sin ωt ) mA

C 380(-sin ωt ) uAD 60.(-sin ωt ) uA


170 cos(ωt ), at 60. Hz− +

2 60 377 d/

2b electricity - ucsd

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