Embed Size (px)
Citation preview
21. Current and Direct Current Circuits21. Current and Direct Current Circuits21-1. Electric Current
Electric current I : Charge flow rate
,tQIav ∆
∆=
dtdQI ≡ (Ampere)
1 A = 1 C/sec.
AIJ ≡
rCurrent Density ; Charge flow rate per unit areaCurrent per area
A electric wireNumber of charge carriers
AtvnVnN d ⋅∆⋅=∆⋅=∆
qAnvtt
qAnvt
qNtQI d
dav ⋅=∆
∆⋅
=∆
⋅∆=
∆∆
=
vd ; Drift velocity
dnqvAIJ ==
dvnqJ rr≡Current density
0vtav +=rr
Emq
mEqvtav cd
rr
rr ττδ =⋅>=<+⋅= 0
0
Em
nqJrr τ2
= E//Jrr
⇒
21-2. Resistance and Ohm’s Law
A
l
Vb VaI
Resistance
IVR ∆
≡ (Ω) Ohm’s Law
1 Ω = 1 V/A
A metal
Ohm’s law : a empirical relation I IRV =∆Not always true
Materials follow Ohm’s law ; Ohmic Materials
∆V
Em
nqJrr τ2
=
lEVVV ab ⋅=−=∆
AJI ⋅= lV
mnq
AI ∆
=τ2
VlA
mnqI ∆=
τ2
1/RσConductivity
Al
AlR ρ
σ==
σρ 1= : Resistivity
τρ 2nq
m=
τρ 2nq
m= T
1∝τ In Metal
)](1[ 00 TT −+= αρρ
T∆∆
=ρ
ρα
0
1
: Temperature coefficient of resistivity
)](1[ 00 TTRR −+= α
At low temperature, ρ → ρ0
Materials
Resistivityρ (Ω·m)
Temperature Coefficientα [(°C)-1]
Silver Copper Gold Aluminum Tungsten Iron Platinum Lead Nichrome Carbon Silicon Glass Quartz
1.59 × 10-8
1.7 × 10-8
2.44 × 10-8
2.82 × 10-8
5.6 × 10-8
10 × 10-8
11 × 10-8
22 × 10-8
1.50 × 10-6
3.5 × 10-8
640 1010 1014
75 × 1016
3.8 × 10-3
3.9 × 10-3
3.4 × 10-3
3.9 × 10-3
4.5 × 10-3
5.0 × 10-3
3.92 × 10-3
3.9 × 10-3
0.4 × 10-3
-0.5 × 10-3
-75 × 10-3
--------
Resistivity and Temperature Coefficient of Resistivity
In Semiconductor
I gE
kT
e2
0
−
= σσ
gEkT
e2
0ρρ =
∆V
In Superconductor
Materials
TemperatureCoefficient α [(°C)-1]
YBa2Cu3O7
Bi-Sr-Ca-Cu-O Tl-Ba-Ca-Cu-O HgBa2Ca2Cu3O8 Nb3Ge Nb3Sn Nb Pb Hg Sn Al Zn
9210512513423.2
21.059.467.184.153.721.190.88
Critical Temperature for Various Superconductors
ρ = 0
• Resistors in Series and in Parallel
AlR ρ= 1−∝⇒ A,lR
In Series
21 RRR +=
∑=i
iRR
In Parallel
21
111RRR +=
∑=i iRR
11
21-5. Electrical Energy and Power
VQU ∆⋅∆=∆
VIVtQ
tU
∆⋅=∆⋅∆∆
=∆
∆
RVRIIRIVIP
22 )(∆
==⋅=∆⋅= (W)
Electrical Power.secJW 11 =
( ) J.secWkWh 63 10633600101 ×=⋅=
Example Lightbulb
100 W Bulb, 24 hrs
Energy = ( ) ( ) kWh.hkW. 422410 =×
21-6. Source of emf
⇒
Battery ⇒ Electromotive force (emf ) εIrVab −= ε Actual voltage depends on the current.
Internal resistanceIRVdc =
dcab VV =
rRI
+=⇒
εIRIr =−εPower :
rIRII 22 +=ε Dissipated energy due to the internal resistance
Let’s neglect the internal resistance!
21-8. Kirchhoff’s Rules and Simple DC Circuitsr
What is the equivalent resistance?
• Kirchhoff’s Rules
1. ∑ 0=a ai
Charge conservationi1
i2
i3
at a junction
0321 =++−=∑ iiii
321 iii +=
0=∆∑closed
V
00
22
11
=∆−=∆−
VRiVRi
i1R1
i2 R2i
∆V
02211 =− RiRi1
2
2
1
RR
ii
=⇒
eqRV
RV
RV
iii∆
=∆
+∆
=
+=
21
21
21
111RRReq
+=⇒
22
11
RViRVi
∆=∆=
2.
Energy conservation
a bI
RI
IRVVV ab −=−=∆
a b
Ra b
ε
IRVVV ab =−=∆
ε=−=∆ ab VVV
a b ε−=−=∆ ab VVVε
iReq 6
ε=r6i
2i
2i
2i ii
i ii
i 2i2i
2i 6i
ε=++ iRiRiR 22
eqiRiR 65 == ε
RReq 65
=
ε
Example 21.9 Applying Kirchhoff’s Rules
a
b
14 V
4Ω
e
c
d
f
10 V 6Ω
2Ω
+ −
− +
I1
I2
I3
321 III =+⇒0=∑ i
( ) ( ) 02610 31 =Ω−Ω− IIV0=∆∑
abcdaV
( ) ( ) 0106144 12 =−Ω+−Ω− VIVI
0=∆∑befcb
V
AI,AI,AI 132 321 −=−==
Example 21.10 A Multiloop Circuit
I1
I3
I2
I3
I1
a
b
5Ω
e
c
d
f
3 V
3Ω
+ −
− +
− +
4 V
8 V
6µF
g
h+ −
I=0
5Ω
321 III =+⇒0=∑ i
( ) ( ) 0534 32 =Ω−Ω− IIV0=∆∑
defcdV
( ) ( ) 0853 12 =+Ω−Ω VII0=∆∑
cfgbcV
A.I,A.I,A.I 0213640381 321 =−==
038 =−∆+− VVV c
0=∆∑abgha
V∆Vc
VVc 11=∆ CVCQ c µ=∆⋅=⇒ 66
21-9. RC Circuits
• Charging a Capacitor
Switch on at t = 0
0=−−ε IRcq
RI ε
=∴ 00==∆⇒ CqVcat t = 0, q = 0
dtdqI =Differential equation at t > 0
)qC(RCRC
qRdt
dq−=−=⇒ εε 1
0=⋅−−ε Rdtdq
cq
RCx
dtdq
dtdx
−=−=⇒qCx −= ε
∫∫ −=t
dtRCx
dx0
1 RCt
exxRCt
xx −
=⇒−= 00
ln
−=
−ε RCt
eCq 1RCt
eCqC −εε =− ⇒
,eCq RCt
−=
−ε 1
τ−−− εεε ==⋅==t
RCt
RCt
eR
eR
eRC
CdtdqI 1 RC=τ
3701 .e =−
Ttq
qIq
Vq
IVRC =
=
=
∆⋅
∆==τ
FC,R µ=Ω= 110 secRC 510−==τ⇒
• Discharging a Capacitor
RdtdqIR
Cq
⋅−==
Iε
S
+
−
+
−
R
C
RCq
dtdq
−=
dtRCq
dq 1−=
tRCq
qln 1
0
−= RCt
eqq −= 0⇒
RCt
RCt
RCt
eI
eRCqe
RCq
dtdqtI
−
−−
=
=
−⋅−=−=
0
00
1)(
QCε
II0
t
t
-I0
