circuit elements. conventional current: widely known as ohm’s law resistance of a long wire:...
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Circuit Elements
Conventional current: Widely known as Ohm’s law
Resistance of a long wire:
Units: Ohm, George Ohm(1789-1854)
Resistance
Resistance combines conductivity and geometry!
Microscopic Macroscopic
Can we write V=IR ?
Microscopic and Macroscopic View
Current flows in response to a DV
L=5 mm
A = 0.002 mm2
Conductivity of Carbon: = 3.104 (A/m2)/(V/m)
What is its resistance R?
(V/A)
What would be the current through this resistor if connected to a 1.5 V battery?
Exercise: Carbon Resistor
Mobility of electrons: depends on temperature
Conductivity and resistance depend on temperature.
Conductivity may also depend on the magnitude of current.
Constant and Varying Conductivity
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!)
Ohmic Resistors
Tungsten: mobility at room temperature is larger than at ‘glowing’ temperature (~3000 K)
V-A dependence:3 V 100 mA1.5 V 80 mA0.05 V 6 mA
R30 19 8
V
I
Is a Light Bulb an Ohmic Resistor?
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
Semiconductors
Capacitors |V|=Q/C, function of time
Batteries: double current, but |V|emf, hardly changes
has limited validity!Ohmic when R is indep-pendent of I!
Conventional symbols:
Nonohmic Circuit Elements
Semiconductors
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
Series Resistance
Know R , find V1,2
Solution:
1) Find current:
2) Find voltage:
3) Check:
Exercise: Voltage Divider
R1
R2
V1
V2
emf
I = I1 + I2 + I3
Parallel Resistance
R1 = 30
R2 = 10
What is the equivalent resistance?
What is the total current?
Alternative way:
Two Light Bulbs in Parallel
What would you expect if one is unscrewed?
Two Light Bulbs in Parallel
A) The single bulb is brighterB) No differenceC) The single bulb is dimmer
Current: charges are moving work is done
Work = change in electric potential energy of charges
Power = work per unit time:
I
Power for any kind of circuit component:
Work and Power in a Circuit
Units:
emf RKnow V, find P
Know I, find P
In practice: need to know P to select right size resistor – capable of dissipating thermal energy created by current.
Power Dissipated by a Resistor
What is the power output of the battery?
Electric field in a capacitor:0
/
AQ
E
E
s
+Q-Q
f
i
ldEV
EsV
sAQ
V0
/
V
s
AQ 0
VQ ~In general:
Definition of capacitance:VCQ
Capacitance
Capacitance of a parallel-plate capacitor:
s
AC 0
Capacitance
Capacitance
Michael Faraday
(1791 - 1867)
VCQ
Units: C/V, Farads (F)
This 1 Farad capacitor is equivalent to a large two-disk capacitor
s=1 mm
D
How large would it be?
s
AC 0
0
CsA
A 1 F 0.001 m
9 10 12 C2 /N m2 A 1.1108 m2
D ~ 10 km (6 miles)
Exercise
Alternative approach:
Energy density:
Energy:
Energy Stored in a Capacitor
Capacitor: Charging and Discharging
Charging Discharging
Positive and negative charges are attracted to each other: how can they leave the plates?
Fringe field is not zero!
How is Discharging Possible?
Electrons in the wire near the negative plate feel a force thatmoves them away from the negative plate.
Electrons near the positive plate are attracted towards it.
Initial moment: brighter?
Will it glow longer?
Parallel Capacitors
Fringe field:R
sAQE
01 2
/
Capacitors in parallel effectively increase A
Will it glow at all?How do electrons flow throughthe bulb?
An Isolated Light Bulb
Why do we show charges nearbulb as - on the left and + onthe right?
Ammeter: measures current I
Voltmeter: measures voltage difference V
Ohmmeter: measures resistance R
Ammeters, Voltmeters and Ohmmeters
0.150
Connecting ammeter:
Conventional current must flow into the ‘+’ terminal and emerge from the ‘-’ terminal to result in positive reading.
Using an Ammeter
Simple commercialammeter
Ammeter Design
Want tiny resistance in coil so current isn’t affected
What happens if not connected correctly?
VAB – add a series resistor to ammeter
R
VI
Measure I and convert to VAB=IR
Connecting Voltmeter:
Higher potential must be connected to the ‘+’ socket and lower one to the ‘-’ socket to result in positive reading.
VoltmeterVoltmeters measure potential difference
R
How would you measure R?
A
Ohmmeter
R
emfI
I
emf
Ohmmeter
Ammeter with a small voltage source
0_ Ctripround VRIemfV
C
QVC
0C
QRIemf
I dQ
dt
emf Q / C
R
Initial situation: Q=0R
emfI 0
Q and I are changing in timedt
d
RC
Q
dt
d
R
emf
dt
d
dt
dIdt
dQ
RCdt
dI 1 I
RCdt
dI 1
Quantitative Analysis of an RC Circuit
IRCdt
dI 1
dtRC
dII
11
tI
I
dtRC
dII 0
11
0
RC
tII 0lnln
RC
t
I
I
0
ln
RC
t
eI
I
0
RCteII /0
Current in an RC circuit
What is I0 ?
RCteR
emfI /
Current in an RC circuit
RC Circuit: Current
What about charge Q?
RCteR
emfI /
Current in an RC circuit
dt
dQI
IdtdQ
t
RCtt
dteR
emfIdtQ
0
/
0
Q C emf 1 e t / RC
RC Circuit: Charge and Voltage
RCteR
emfI /
Current in an RC circuit
Charge in an RC circuit
Q C emf 1 e t / RC
Voltage in an RC circuit
V emf 1 e t / RC
RC Circuit: Summary
RCteR
emfI /
Current in an RC circuit
When time t = RC, the current I drops by a factor of e.
RC is the ‘time constant’ of an RC circuit.
37.0718.2
11/ ee RCt
The RC Time Constant
A rough measurement of how long it takes to reach final equilibrium
What is the value of RC?
About 9 seconds
Question
A 0.5 farad capacitor isconnected to a 1.5 voltbattery and a bulb, andcurrent runs until the bulbgoes out. What is theabsolute value of thecharge on one plate of thecapacitor?
A) 0.33 CB) 0.5 CC) 0.75 CD) 1.5 CE) 3.0 C
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