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1
Chapter 20
Circuits
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1) Electric current and emf
a) Potential difference and charge flowBattery produces potential difference causing flow of charge in conductor
3
b) Current: I = q/t
∆ q is charge that passes the surface in time ∆ t
Units: C/s = ampere = A
4
• Drift velocity: average velocity of electrons
~ mm/s
• Signal velocity: speed of electric field
= speed of light in the material ~108 m/s
5
• emf = electromotive force = maximum potential difference produced by a device
• Symbol: E• emf is not a force, but it causes current to flow
Eis like gh
c) Electromotive force, emf
gravitational analogy for a circuit
battery
6
• Symbol for a perfect seat of emf
E
V = E
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• Real battery
r
R
V < Ein general
Battery terminals
E
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2) Ohm’s Law
• Ohm’s law: for some devices (conductors), I is proportional to V:
IV Device
I
V
V = IR
• R = Resistance = proportionality constant = V/I
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• Current depends on voltage
IV Device
I
V
I
V V
I
and on the device
• Resistance R = V / I, not necessarily constant
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• Ohmic material obeys Ohm’s Law: R is constant• R is a property of the device
IV Device
• symbol:
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3) Resistivity
• Property of material; zero for superconductors
• For cylindrical conductor:
• R is proportional to L• R is proportional to 1/A• R is proportional to L / A• Define resistivity as the proportionality constant
€
R = ρL
A
a) Definition
A
L
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b) values
• Conductors: ~ 10-8 m (Cu, Ag best)
• Semiconductors: ~ 1 - 103 m (Ge, Si)
• Insulators: ~ 1011 - 1016 m (rubber, mica)
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c) Temperature dependence
• Resistivity is linear with temperature:
For metals, > 0 (resistance increases with temp)For semiconductors, < 0 (resistance decreases)
€
=a + bT
€
=0 + b(T −T 0)
€
0 = resistivity at T = T0
€
/ρ0 = 1 + α (T −T 0)€
a + bT0
€
= coefficient of resistivity (C º -1 )
€
=0(1+ α (T −T 0))
€
⇒ R = R0(1 + α (T −T 0))
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d) Superconductors• Below critical temp Tc, –> 0
– Current flows in loop indefinitely– Quantum transitions not possible
Tc typically < 10 K, but can be > ~ 75 K (high Tc ceramics) (record is 138 K)
Applications: MRI, MagLev trains
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4) Power and Energy
• Energy lost or gained by q is UqV
• Power:
V I
€
P =ΔU
Δt
Units: (C/s)(J/C) = J/s = WConsumed energy = P t: [kW h] = (1000 W) (3600 s) = 3.6 MJ
€
=qV
Δt
€
P = VI
a) Power dissipated in a device
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b) Power dissipated in resistors
V I V = IR
€
P = VI
€
=(IR)I
€
⇒
€
P = I2R
€
P = VI
€
=VV
R
€
⇒
€
P =V 2
R
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6) AC/DC
a) Direct (Constant) Current
IV
V
t
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b) Alternating Current
VQuickTime™ and a
TIFF (Uncompressed) decompressorare needed to see this picture.
It
V0
-V0
V
ac generator alternates polarity:
€
e.g. V = V0 sin(ωt)
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Average voltage: zero
€
Vrms = V 2 =V0
2t
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
V0
-V0
V
t
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
I0
-I0
I
Average current: zero
€
Irms = I2 =I0
2
Average power:
€
P = 12 V0I0
For resistors
€
=VrmsIrms
€
P =V0
2
I0
2
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6) Circuit wiring
a) Basic circuit
IE
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b) Ground
One point may be referred to as ground
IE
The ground may be connected to “true” ground through water pipes, for example.
IE=
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d) Open circuit
EI
c) Short circuit
E
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f) Parallel connection
e) Series connection
same currentI
same voltageV
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7) Resistors in series
For perfect conductors
€
V = V1 + V2
From Ohm’s law
€
V1 = IR1 and V2 = IR2
€
So, V = IR1 + IR2
€
=I(R1 + R2)
€
Or, V = IRS
€
RS = R1 + R2if
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In general, for series resistors,
€
RS = R1 + R2 + R3 +L
€
RS = Rii
∑
Find the current and the power through each resistor.
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Voltage divider
€
I =V
RS
=V
R1 + R2
V=10VR1=6
R2=4
€
=1A
Current is the same in both resistors
I
Voltages divide in proportion to R
€
V1 = IR1 = 6V
€
V2 = IR2 = 4VVo
Output Voltage:
€
Vo = IR2
€
=VR2
R1 + R2
⎛
⎝ ⎜ ⎞
⎠ ⎟
€
=V
R1 + R2
R2
V
Vo
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8) Resistors in parallel
a) General caseConservation of charge
€
I = I1 + I2
Ohm’s Law
€
V = I1R1 and V = I2R2
€
So, I =V
R1
+V
R2
€
=V1
R1
+1
R2
⎛
⎝ ⎜ ⎞
⎠ ⎟
€
Or, V = IRP
€
1
RP
=1
R1
+1
R2
if
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€
1
RP
=1
R1
+1
R2
€
RP =R1R2
R1 + R2
€
=R1 //R2
• Equivalent resistance is smaller than either R1 or R2
• Conductance adds
In general, for parallel resistors,
€
1
RP
=1
R1
+1
R2
+1
R3
+L
or
€
1
RP
=1
R ii∑
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conductance adds
30
parallel connections in the home
31
b) Special cases
i) Equal resistance
€
RP = R //R =R2
2R
€
=R
2
ii) Very unequal resistors (e.g. 1 and 1 M
€
RP = R1 //R2 =R1R2
R1 + R2
€
=(1)(106)Ω
1 +106 ≅ 1Ω
€
If R2 >> R1, then R1 + R2 ≅ R2
€
so RP ≅R1R2
R2
= R1 RP = the smaller value