1. A Microscopic View of Electric Circuits
Chapter 19
A Microscopic View of Electric Circuits

2. Current in a Circuit A microscopic view of electric circuits:
Are charges used up in a circuit?
How is it possible to create and maintain a nonzero electric field inside a wire?
What is the role of the battery in a circuit?
In an electric circuit the system does not reach equilibrium!
Steady state and static equilibrium
Static equilibrium:
no charges are moving
Steady state (Dynamic Equilibrium):
charges are moving
their velocities at any location do not change with time
no change in the deposits of excess charge anywhere

3. Current in Different Parts of a Circuit
What happens to the charges that flow through the circuit?
Is the current the same in all parts of a series circuit?
What would be IA compared to IB?
IB = 0 all electrons are used up in the bulb
IB < IA some of the electrons are used up in the bulb
IB = IA

4. Current in Different Parts of a Circuit
IB = 0 all electrons are used up in the bulb
electrons disappear in the light bulb
requires nuclear reaction
b) electrons accumulate in the light bulb
Electrons – can they be used up? – current is just a flow of electrons!
Annihilation – only way to destroy them – a lot of energy – there is no nuclear reaction in light bulb, need antimatter, besides will see a lot of energy.
will notice charge
2) IB < IA some of the electrons are used up in the bulb
3) IB = IA

5. Current in Different Parts of a Circuit
3) IB = IA
Test:
1. Can use compass needle deflection for wire A and B
2. Run wires A and B together above compass
Get double the deflection with two wires. A B

6. Current in Different Parts of a Circuit
IB = IA in a steady state circuit
We cannot get something for nothing!
What is used up in the light bulb?
Energy is transformed from one form to another
Electric field – accelerates electron
Friction – energy is lost to heat
Battery – chemical energy is used up
Rub your hands – what happens? They heat up. Are they used up? No!
Closed circuit – energy losses to heat:
not an isolated system!

7. Gustav Robert Kirchhoff
Current at a Node
i1 = i2
i2 = i3 + i4
The current node rule
(Kirchhoff node or junction rule [law #1]):
In the steady state, the electron current entering a node in a circuit is equal to the electron current leaving that node
Gustav Robert Kirchhoff
( )
(consequence of conservation of charge)
Can also be expressed in terms of conventional current
Also called Kirchhoff's first law, Kirchhoff's point rule, Kirchhoff's junction rule, and Kirchhoff's first rule.The principle of conservation of electric charge implies that:
At any point in an electrical circuit where charge density is not changing in time, the sum of currents flowing towards that point is equal to the sum of currents flowing away from that point.

8. Exercise Write the node equation for this circuit.
What is the value of I2?
I1 + I4 = I2 + I3
I2 = I1 + I4 - I3 = 3A
What is the value of I2 if I4 is 1A?
I1 + I4 = I2 + I3
Can also be expressed in terms of conventional current
I2 = I1 + I4 - I3 = -2A
Charge conservation: 1A
Ii > 0 for incoming
Ii < 0 for outgoing

9. Exercise Is the current everywhere the same? i1 = i2 + i3
Make complex circuit, one wire in, one out, ask what I is – circle all elements so that only one wire goes in, one out – charge conservation.
i1 = i2 + i3

10. Motion of Electrons in a Wire
In a current-carrying wire, the electric field drives the mobile charges.
What is the relationship between the electric field and the current?
Why is an electric field required?
Electrons lose energy to the lattice of atomic cores. Electric field must be present to increase the momentum of the mobile electrons.
Collisions with Core gives frictional force.
In steady state, net force is zero.

11. The Drude Model E Momentum principle: Speed of the electron:
Precise picture – need quanum mechanics. But simple classical Drude model allows us to unterstand most of the important aspects of circuits on a microscopic level.
Mobile electrons move under the influence of the electric field inside the metal
ELECTRIC FIELD IN METAL
Average ‘drift’ speed:
- average time between
collisions

12. The Drude Model Average ‘drift’ speed: - average time between
collisions
For constant temperature
Paul Drude
( )
u – mobility of an electron
Electron current:

13. Typical E in a Wire
Drift speed in a copper wire in a typical circuit is m/s.
The mobility is u= (m/s)/(N/C). Calculate E.
The drift speed was calculated in one of the earlier exercises. If we use the relation between mobility and mean time between collisions, we find that deltaT is about seconds.
Electric field in a wire in a typical circuit is very small

14. Mean Time Between Collisions
u= (m/s)/(N/C).
We find that In this time, electron travels only
1.5x10-18 m due to drift velocity.
But electron also has a random velocity of about 105 m/s. (To understand this, one must use Quantum Mechanics!)
Thus, in 3x10-14 s, electron actually travels 3x10-9 m or about 30 Angstroms - more than 10 lattice spacings.

15. E and Drift Speed
In steady state current is the same everywhere in a series circuit.
Ethick
Ethin i i
What is the drift speed?
Electron current depends on cross section of wire
Since the drift speed must be greater in the thinner wire, the electric field in the thinner wire must be larger
If R_thin is twice smaller than R_thick?
Highway traffic – in cogestion density changes, speed may stay the same
Note: density of electrons n cannot change if same metal
What is E?

16. Exercise 0.1 mm 1 mm vthick = 410-5 m/s vthin = ? Ethin= 10-1 N/C
Ethick= ?
Radius is a factor of 10 different -> v is 100 times larger
vthin = 40010-5 m/s
Ethick= 10-3 N/C

17. Question
1.5.n1 = n2
2.u1 = u2
1018
1 mm
2 mm
Every second 1018 electrons enter the thick wire.
How many electrons exit from the thin wire every second?
Same number of electrons/second must exit thin wire.
C) 2 x 1018
D) 4 x 1018
E) 12 x 1018
1018
1.5 x 1018

18. Question
1.5.n1 = n2
2.u1 = u2
1018
1 mm
2 mm
What is the ratio of the electric field, E1/E2?
D. Same number of electrons/second must exit thin wire.
Ratio of electric field in thin wire to thick wire: E1/E2 = n2*A2*u2/(n1*A1*u1) = 1.5*4*2=12
3:1
6:1
8:1
12:1

19. Direction of Electric Field in a Wire
E must be parallel to the wire
E is the same along the wire
Does current fill the wire?
Is E uniform across the wire?
B field measurements are the same all along the wire (unless we encounter loops).
VAB
VCD

20. Electric Field in a Wire
What charges make the electric field in the wires? E
Bulb filament and wires are metals –
there cannot be excess charges in the interior
Are excess charges on the battery? E
There must be an electric field in the wires that is responsible for moving the charge carriers around the circuit.
The magnitude of E must be the same throughout a wire of uniform cross section and material.
The wire itself is electrically neutral, so there must be excess charges that give rise to the E-field.
If the excess charges were on the poles of the battery, we might expect the field to be greater closer to the battery. Moving the bulb closer to the battery does not change the current (as indicated by the brightness of the bulb). Also, changing the orientation of the bulb has no effect. If the excess charges responsible for the current resided on the battery poles, we would expect a directionality effect. E

21. A Mechanical Battery Van de Graaff generator Electron Current
Use because it is easier to understand than chemical battery.
Electron Current

22. Field due to the Battery
Blue = vdrift
Green arrows represent E due only to charges in battery. Blue arrows represent drift velocity direction. As long as the field still points to the right at location 4, positive charge continues to build up. This will continue until the field points to the left. Eventually, the field at locations 3, 4 and 5 must have the same magnitude so that the electron current is the same everywhere.
Ebends
In the steady state there must be some other charges somewhere that contribute to the net electric field in such a way that the electric field points upstream everywhere.

23. Field due to the Battery
If this were not the case, ie, if E varied along the length of the wire (of uniform area, material), we could not conserve charge. If the surface charge were distributed uniformly on the wire surface, current would not flow (no particular direction). Thus, it must not be distributed uniformly!
Surface charge arranges itself in such a way as to produce a pattern of electric field that follows the direction of the wire and has such a magnitude that current is the same along the wire.

24. Field due to Battery E
Smooth transition from + surface charge to – to provide
constant E.
Greater density of surface charge on right end of cylinder (representing the wire) than at left end. Hence, field along axis points to the right.
The amount of surface charge is proportional to the voltage.

25. Connecting a Circuit
When making the final connection in a circuit, feedback forces a rapid rearrangement of the surface charges leading to the steady state.
This period of adjustment before establishing the steady state is called the initial transient.
In book: Nichrome wire – explain that it has lower mobility than copper, so current is small and battery is not short circuited.
It would happen in any wire.
The initial transient

26. Connecting a Circuit Static equilibrium: nothing moving (no current)
Initial transient: short-time process
leading to the steady state
In steady state, surface charges rearrange themselves so that the E field points along the wire and is the same magnitude all along the wire.
3. Steady state: constant current
(nonzero)

27. Why Light Comes on Right Away?
How fast electrons move?
v ~ 50 micron/s
It may take hours before electrons from the battery reach light bulb!
AC current: E changes direction back and forth ‘shaking’ electrons
The light comes on right away because the rearrangement of surface charges in the circuit takes place at the speed of light. The final steady state is established in a few nanoseconds, after which the electron sea circulates slowly at the drift velocity around the circuit.

28. Feedback
Feedback: produces surface charge of the right amount to create the appropriate steady-state electric field and maintain it.
More incoming than outgoing electron current - > excess charge on surface builds up

29. Feedback i Extra charge on bend Feedback makes current follow the wire
There must be some excess charge on the outside of the wire, all along the wire.
Through the feedback mechanism this surface charge will automatically arrange itself so that the net force inside the wire always points along the direction of the wire.
Feedback makes current follow the wire

30. Surface Charge and Resistors
Just after connection:
E may be the same
everywhere
After steady state is reached:
Just after connecting the circuit but before equilibrium is established, the electric field in the narrow resistor might temporarily be about the same as the electric field in the thick wires, leading to a comparable drift velocity. But since Athin is less than Athick, electrons accumulate at the right end of the thin wire, increasing the E field in the thin wire. Similarly, a deficiency of electrons develops at the left end because the number of electrons emerging from the wire is too small. These fields reach equilibrium when as many electrons leave the thin wire as enter. This happens when Ethin = Athick/Athin * Ethick.

31. Surface Charge Distribution in a Circuit
1. Current must be constant: based on
draw distribution of E
2. Based on E draw approximate distribution of surface charges

32. Exercise
What is wrong? See 18.X.10

33. A Wide Resistor low mobility
In the transient electrons pile up not only on the surface BUT also at the interface

34. Current at a Node i1 = i2 i2 = i3 + i4 The current node rule
(Kirchhoff node or junction rule [law #1]):
In the steady state, the electron current entering a node in a circuit is equal to the electron current leaving that node
Can also be expressed in terms of conventional current

35. Energy in a Circuit Vwire = EL Vbattery = ?
From charge conservation and definition of steady state: Node rule – tells us about relations between currents in different parts. But what about actual current? How large is it.
To answer – turn to energy conservation principle. Consider one electron as it is transported around the circuit.
Energy gained by e as it is transported between terminals in a battery is dissipated in the rest of the circuit.
Use round-trip.
Energy conservation (the Kirchhoff loop rule [2nd law]):
V1 + V2 + V3 + … = along any closed path in a circuit
V= U/q  energy per unit charge

36. Potential Difference Across the Battery
Coulomb force on each e FC
non-Coulomb force on each e
1. FC =eEC EC
2. FC =FNC
Fully charged battery.
Energy input per unit charge
emf – electromotive force
Fc = Fnc in equilibrium.
The function of a battery is to produce and maintain a charge separation.
The emf is measured in Volts, but it is not a potential difference!
The emf is the energy input per unit charge.
chemical, nuclear, gravitational…

37. Chemical Battery
The first battery was created by Alessandro Volta in To create his battery, he made a stack by alternating layers of zinc, blotting paper soaked in salt water and silver.
You can create your own voltaic pile using coins and paper towel. Mix salt with water (as much salt as the water will hold) and soak the paper towel in this brine. Then create a pile by alternating pennies and nickels. See what kind of voltage and current the pile produces.
Probably the simplest battery you can create is called a zinc/carbon battery. By understanding the chemical reaction going on inside this battery you can understand how batteries work in general. Imagine that you have a jar of sulfuric acid (H2SO4). Stick a zinc rod in it, and the acid will immediately start to eat away at the zinc. You will see hydrogen gas bubbles forming on the zinc, and the rod and acid will start to heat up. Here's what is happening:
The acid molecules break up into three ions: two H+ ions and one SO4-- ion.
The zinc atoms on the surface of the zinc rod lose two electrons (2e-) to become Zn++ ions.
The Zn++ ions combine with the SO4-- ion to create ZnSO4, which dissolves in the acid.
The electrons from the zinc atoms combine with the hydrogen ions in the acid to create H2 molecules (hydrogen gas). We see the hydrogen gas as bubbles forming on the zinc rod.
If you now stick a carbon rod in the acid, the acid does nothing to it. But if you connect a wire between the zinc rod and the carbon rod, two things change:
The electrons flow through the wire and combine with hydrogen on the carbon rod, so hydrogen gas begins bubbling off the carbon rod.
There is less heat. You can power a light bulb or similar load using the electrons flowing through the wire, and you can measure a voltage and current in the wire. Some of the heat energy is turned into electron motion.
The electrons go to the trouble to move to the carbon rod because they find it easier to combine with hydrogen there. There is a characteristic voltage in the cell of 0.76 volts. Eventually, the zinc rod dissolves completely or the hydrogen ions in the acid get used up and the battery "dies."
A battery maintains a potential difference across the battery.

38. Field and Current in a Simple Circuit
Round-trip potential difference:
Field in Battery from + to -
Field in wire : opposite direction
Neglect battery’s internal resistance in this example.
Start at negative plate and go counter-clockwise
We will neglect the battery’s internal resistance for the time being.

39. Field and Current in a Simple Circuit
Round-trip potential difference:
Path 1
Path 2

40. V Across Connecting Wires
The number or length of the connecting wires has little effect on the amount of current in the circuit.
uwires >> ufilament 
Differences in length tend to be less than factors of 10 while differences in mobility can be factors of 100 to We’ve neglected the battery’s internal resistance.
Write second line in terms of L/(nAu) for the wire and for the filament.
Work done by a battery goes mostly into energy dissipation in the bulb (heat).

41. General Use of the Loop Rule
V1 + V2 + V3 + V4 = 0
Add pics of coffeemakers, refrigerators etc. as elements in a circuit.
(VB-VA)+ (VC-VB)+ (VF-VC)+ (VA-VF)=0

42. Application: Energy Conservation
V1 + V2 + V3 + Vbatt = 0
(-E1L1) + (-E2L2) + (-E3L3) + emf = 0
An electron traversing the circuit D-C-B-A suffers a decrease in its electric potential energy. Where does this loss of energy appear?
Approximately 1.5 volts
Not enough information to determine the current.
Note that the field in the thin wire is much greater than that in the thick wire, as expected.
Ask question about area under the E-plot.

43. E2>E1, therefore vdrift1 < vdrift2 - c

44. I1=i2 n1*E1*u1*A1 = n2*E2*u2*A2 E1/E2 = n2*u2/(n1*u1) = 3 u2/u1

45. C

46. Twice the Length Nichrome wire (resistive)
Quantitative measurement of current with a compass
Electric field is half -> half the drift speed -> half the current. Note that we are considering the wire to be a “resistive” wire.
Current is halved when increasing the length of the wire by a
factor of 2.

47. Doubling the Cross-Sectional Area
Loop: emf - EL = 0
Electron current in the wire increases by a factor of two if the cross-sectional area of the wire doubles.
Electric field in the wire doesn’t change when we use a thicker wire -> current doubles
Nichrome wire

48. Two Identical Light Bulbs in Series
Two identical light bulbs in series are the same as one light bulb with twice as long a filament.
Identical light bulbs
The two bulbs emit “redder” light than the single bulb, indicating that the filaments are not has hot. Here, we consider delta_V to represent the fixed emf of the battery.
Electron mobility in metals decreases as temperature increases!
Conversely, electron mobility in metals increases as temperature
decreases. Thus, the current in the 2-bulb circuit is slightly
more than that in the one-bulb circuit.

49. Two Light Bulbs in Parallel
L … length of bulb filament
1. Path ABDFA:
2. Path ACDFA: F
iB = iC
ibatt = 2iB
3. Path ABDCA:
What happens if one light bulb is removed?
We can think of the two bulbs in parallel as equivalent to increasing the cross-sectional area of one of the bulb filaments.

50. Long and Round Bulb in Series
only round bulb:
Why round bulb does not glow? The calculation in the box shows the electric field required to light the round bulb.
The electric field doesn’t produce sufficient current through the filament to make it glow.
The round bulb doesn’t light up!

51. Two Batteries in Series
Why light bulb is brighter with two batteries?
Two batteries in series can drive more current:
Potential difference across two batteries in series is 2emf  doubles electric field everywhere in the circuit  doubles drift speed  doubles current.
Work per second:
After getting i for both – ask how much brighter? How much more heat dissipate with 2 batteries?
In reality: not 4x because mobility changes with temperature., I.e. current also does not go twice up.

52. Analysis of Circuits The current node rule (Charge conservation)
Kirchhoff node or junction rule [1st law]:
In the steady state, the electron current entering a node in a circuit is equal to the electron current leaving that node
Electron current: i = nAuE
Conventional current: I = |q|nAuE
The loop rule (Energy conservation)
Kirchhoff loop rule [2nd law]:
V1 + V2 + V3 + … = along any closed path in a circuit
V= U/q  energy per unit charge

53. Resistance Widely known as Conventional current: Ohm’s law
George Ohm
( )
George Ohm, studied resistance in early 1800’s
R – combines geometry and material properties
We have not developed any new physical principles – just expressed the I in terms of R and dV.
His father educated him at home.
In 1805 Ohm entered the University of Erlangen but he became rather carried away with student life. Rather than concentrate on his studies he spent much time dancing, ice skating and playing billiards. Ohm's father, angry that his son was wasting the educational opportunity that he himself had never been fortunate enough to experience, demanded that Ohm leave the university after three semesters. Ohm went (or more accurately, was sent) to Switzerland where, in September 1806, he took up a post as a mathematics teacher in a school in Gottstadt bei Nydau.
His private studies had stood him in good stead for he received a doctorate from Erlangen on 25 October 1811 and immediately joined the staff as a mathematics lecturer. After three semesters Ohm gave up his university post. He could not see how he could attain a better status at Erlangen as prospects there were poor while he essentially lived in poverty in the lecturing post. The Bavarian government offered him a post as a teacher of mathematics and physics at a poor quality school in Bamberg and he took up the post there in January 1813.
This was not the successful career envisaged by Ohm and he decided that he would have to show that he was worth much more than a teacher in a poor school. He worked on writing an elementary book on the teaching of geometry while remaining desperately unhappy in his job. After Ohm had endured the school for three years it was closed down in February The Bavarian government then sent him to an overcrowded school in Bamberg to help out with the mathematics teaching.
On 11 September 1817 Ohm received an offer of the post of teacher of mathematics and physics at the Jesuit Gymnasium of Cologne. This was a better school than any that Ohm had taught in previously and it had a well equipped physics laboratory. As he had done for so much of his life, Ohm continued his private studies reading the texts of the leading French mathematicians Lagrange, Legendre, Laplace, Biot and Poisson. He moved on to reading the works of Fourier and Fresnel and he began his own experimental work in the school physics laboratory after he had learnt of Oersted's discovery of electromagnetism in At first his experiments were conducted for his own educational benefit as were the private studies he made of the works of the leading mathematicians.
The Jesuit Gymnasium of Cologne failed to continue to keep up the high standards that it had when Ohm began to work there so, by 1825, he decided that he would try again to attain the job he really wanted, namely a post in a university. Realising that the way into such a post would have to be through research publications, he changed his attitude towards the experimental work he was undertaking and began to systematically work towards the publication of his results [1]:-
Overburdened with students, finding little appreciation for his conscientious efforts, and realising that he would never marry, he turned to science both to prove himself to the world and to have something solid on which to base his petition for a position in a more stimulating environment.
In fact he had already convinced himself of the truth of what we call today "Ohm's law" namely the relationship that the current through most materials is directly proportional to the potential difference applied across the material. The result was not contained in Ohm's firsts paper published in 1825, however, for this paper examines the decrease in the electromagnetic force produced by a wire as the length of the wire increased. The paper deduced mathematical relationships based purely on the experimental evidence that Ohm had tabulated.
In two important papers in 1826, Ohm gave a mathematical description of conduction in circuits modelled on Fourier's study of heat conduction. These papers continue Ohm's deduction of results from experimental evidence and, particularly in the second, he was able to propose laws which went a long way to explaining results of others working on galvanic electricity. The second paper certainly is the first step in a comprehensive theory which Ohm was able to give in his famous book published in the following year.
What is now known as Ohm's law appears in this famous book Die galvanische Kette, mathematisch bearbeitet (1827) in which he gave his complete theory of electricity. The book begins with the mathematical background necessary for an understanding of the rest of the work. We should remark here that such a mathematical background was necessary for even the leading German physicists to understand the work, for the emphasis at this time was on a non-mathematical approach to physics. We should also remark that, despite Ohm's attempts in this introduction, he was not really successful in convincing the older German physicists that the mathematical approach was the right one. To some extent, as Caneva explains in [1], this was Ohm's own fault:-
... in neither the introduction nor the body of the work, which contained the more rigorous development of the theory, did Ohm bring decisively home either the underlying unity of the whole or the connections between fundamental assumptions and major deductions. For example, although his theory was conceived as a strict deductive system based on three fundamental laws, he nowhere indicated precisely which of their several mathematical and verbal expressions he wished to be taken as the canonical form.
It is interesting that Ohm's presents his theory as one of contiguous action, a theory which opposed the concept of action at a distance. Ohm believed that the communication of electricity occurred between "contiguous particles" which is the term Ohm himself uses. The paper [8] is concerned with this idea, and in particular with illustrating the differences in scientific approach between Ohm and that of Fourier and Navier. A detailed study of the conceptual framework used by Ohm in formulating Ohm's law is given in [6].
As we described above, Ohm was at the Jesuit Gymnasium of Cologne when he began his important publications in He was given a year off work in which to concentrate on his research beginning in August 1826 and although he only received the less than generous offer of half pay, he was able to spend the year in Berlin working on his publications. Ohm had believed that his publications would lead to his receiving an offer of a university post before having to return to Cologne but by the time he was due to begin teaching again in September 1827 he was still without such an offer.
Although Ohm's work strongly influenced theory, it was received with little enthusiasm. Ohm's feeling were hurt, he decided to remain in Berlin and, in March 1828, he formally resigned his position at Cologne. He took some temporary work teaching mathematics in schools in Berlin.
He accepted a position at Nüremberg in 1833 and although this gave him the title of professor, it was still not the university post for which he had strived all his life. His work was eventually recognised by the Royal Society with its award of the Copley Medal in He became a foreign member of the Royal Society in Other academies such as those in Berlin and Turin elected him a corresponding member, and in 1845 he became a full member of the Bavarian Academy.
This belated recognition was welcome but there remains the question of why someone who today is a household name for his important contribution struggled for so long to gain acknowledgement. This may have no simple explanation but rather be the result of a number of different contributary factors. One factor may have been the inwardness of Ohm's character while another was certainly his mathematical approach to topics which at that time were studied in his country a non-mathematical way. There was undoubtedly also personal disputes with the men in power which did Ohm no good at all. He certainly did not find favour with Johannes Schultz who was an influential figure in the ministry of education in Berlin, and with Georg Friedrich Pohl, a professor of physics in that city.
Electricity was not the only topic on which Ohm undertook research, and not the only topic in which he ended up in controversy. In 1843 he stated the fundamental principle of physiological acoustics, concerned with the way in which one hears combination tones. However the assumptions which he made in his mathematical derivation were not totally justified and this resulted in a bitter dispute with the physicist August Seebeck. He succeeded in discrediting Ohm's hypothesis and Ohm had to acknowledge his error. See [10] for details of the dispute between Ohm and Seebeck.
In 1849 Ohm took up a post in Munich as curator of the Bavarian Academy's physical cabinet and began to lecture at the University of Munich. Only in 1852, two years before his death, did Ohm achieve his lifelong ambition of being appointed to the chair of physics at the University of Munich.
Resistance of a long wire:
Units: Ohm, 
Resistance combines conductivity and geometry!

54. Microscopic and Macroscopic View
Microscopic Macroscopic
Can we write V=IR ?
No – potential has no meaning in this contents, only potential difference.
V is the potential at a particular location
Current flows in response to a DV

55. Exercise: Carbon Resistor
A = mm2
Conductivity of Carbon:
 = (A/m2)/(V/m)
L=5 mm
What is its resistance R?
(V/A)
What would be the current through this resistor if connected to a 1.5 V battery?

56. Constant and Varying Conductivity
Mobility of electrons: depends on temperature
Conductivity and resistance depend on temperature.
Conductivity may also depend on the magnitude of current.

57. Ohmic Resistors Ohmic resistor: resistor made of ohmic material
Ohmic materials: materials in which conductivity  is independent of the amount of current flowing through
not a function of current
Examples of ohmic materials:
metal, carbon (at constant T!)

58. Is a Light Bulb an Ohmic Resistor?
Tungsten: mobility at room temperature is larger than at ‘glowing’ temperature (~3000 K)
V-A dependence:
3 V 100 mA
1.5 V 80 mA
0.05 V mA R
30 
19 
8  V I
Last slide shown on 10/18.

59. Semiconductors
Metals, mobile electrons: slightest V produces current.
If electrons were bound – we would need to apply some field to free some of them in order for current to flow. Metals do not behave like this!
Semiconductors: n depends exponentially on E
Conductivity depends exponentially on E
Conductivity rises (resistance drops)
with rising temperature

60. Nonohmic Circuit Elements
Semiconductors
|V|=Q/C, function of time
Capacitors
Batteries: double current, but |V|emf, hardly changes
has limited validity!
Ohmic when R is indep-
pendent of I!
Conventional symbols:

61. Series Resistance Vbatt + V1 + V2 + V3 = 0
emf - R1I - R2I - R3I = 0
emf = R1I + R2I + R3I
emf = (R1 + R2 + R3) I
emf = Requivalent I , where Requivalent = R1 + R2 + R3

62. Exercise A certain ohmic resistor has a resistance of 40 .
A second resistor is made of the same material, but is three times longer and has a half of the cross-sectional area.
What is its resistance?
Resistor 1:
Resistor 2:
What would be an equivalent resistance of these two resistors in series?

63. Exercise: Voltage Divider
Know R , find V1,2 R1 R2
V1
V2
emf
Solution:
1) Find current:
2) Find voltage:
3) Check:

64. Parallel Resistance I = I1 + I2 + I3
Req is less than the individual R’s.

65. Two Light Bulbs in Parallel
What is the equivalent resistance?
R2 = 10 
What is the total current?
Alternative way:

66. Two Light Bulbs in Parallel
What would you expect if one is unscrewed?
Unscrew one bulb – what happens with brightness?
With 2 bulbs, equivalent R = R/2, so current with 2 bulbs is double that with one. But current divides equally between the two paths, B and C. Thus, each bulb sees same current as it would in a single bulb circuit. (Sees same 2*emf.)
The single bulb is brighter
No difference
The single bulb is dimmer

67. Work and Power in a Circuit
Current: charges are moving  work is done
Work = change in electric potential energy of charges
Power = work per unit time: I
I flowing thru the component, delta V across that same component.
Power for any kind of circuit component:
Units:

68. Example: Power of a Light Bulb
I = 0.3 A
Units:
emf = 3V

69. Power Dissipated by a Resistor
emf R
Know V, find P
Know I, find P
What is R in previous slide?
In practice: need to know P to select right size resistor – capable of dissipating thermal energy created by current.
What is the power output of the battery?

70. Energy Stored in a Capacitor
Alternative approach:
Energy density:
dW = work done to add a small amount of charge. Q = charge already on capacitor when we add an additional amount dQ.
Amount of work required to charge the capacitor = energy stored in the capacitor.
Ignores fringe field since this is very small compared to interior field.
Cannot use energy density forumula if capacitor is filled with dielectric!
Energy:

71. Question
A certain capacitor with only air between its plates has capacitance
C and is connected to a battery for a long time until the potential
difference across the capacitor is equal to 3 V. The battery is then
removed from the circuit and a dielectric (K=2) is inserted between
the capacitor plates filling the entire volume. The energy stored in
the capacitor WITH dielectric compared to the energy stored
Without dielectric is:
The same
Larger by a factor of 2
Smaller by a factor of 2
D) Larger by a factor of 4
E) Smaller by a factor of 4
C-->KC, energy= 1/2 Q**2/C since charge is fixed. So correct answer is C. Other way: 1/2 C*V**2. C-->KC, V-->V/2