Experimental Physics IIa - Electric current and circuits 1
Experimental Physics EP2a
Electricity and Thermodynamics
– Electric current –Ohm‘s law, power, circuits, Kirhoff‘s rules
https://bloch.physgeo.uni-leipzig.de/amr/
Experimental Physics IIa - Electric current and circuits 2
Electric current
The direction of current is conventionally assumed to be opposite to the direction of flow of electrons.
+
++
tQIav D
D=
dtdQI = [ ]A
sC
=úûù
êëé
xD
Aq
tvdD
tQIav D
D= d
d nqAvttnqAv=
DD
=
26 m103 -´=A 3g/cm95.8=cr
g/mol5.63=cM A10=I
m/s102 4-´»dv
Experimental Physics IIa - Electric current and circuits 3
Drude’s theory of electric conduction
-
-
E!
EqF!!
= ame!
=
tavv if!!!
+= tmEqvi
!!+=
tmEqvd
!!= average time
between collisions
dvnqJ !!= EJ
!!s=
temEnqJ!
!2=
vl
=t
Experimental Physics IIa - Electric current and circuits 4
Ohm’s law
dvnqAIJ !!
!=º EJ
!!s=
s - conductivity
Materials obeying this equation are ohmic.
lEV ×=DslJ ×
= IAls
=
sAlR º resistance [ ]W
- + - +lAR=º
sr 1
resistivity
Experimental Physics IIa - Electric current and circuits 5
Current-potential curve
Normal resistors
Rectifying resistor
Experimental Physics II - Electric circuits 6
Power------------
++++++++++++
-
VqW D=
tQV
tU
DDD
=DD
VID=
( )RVRIP
22 D
==
-Joule heating
Q
QD
1V 2V
Experimental Physics IIa - Electric current and circuits 7
Temperature coefficient of resistivity
( )( )00 1 TT -+= arrdTdr
ra
0
1=! = # $%
Experimental Physics II - Electric circuits 8
Electromotive force
The work per unit charge is called the emf of the source.
+ -eterminal voltage
V
I
e
R
IrV -=D e
+ -e
R
r
VD
IRV =D
rRI
+=e
I
Experimental Physics II - Electric circuits 9
+ -e
R
r
VD
Maximal power delivered
rRI
+=e
RIP 2= ( )22
rRR
+=e
0 5 10 15 20 25 30 356
8
10
12
14
16
18
20
22
24
26
Pow
er, P
Resistance, R
( ) ( )02 32
22=
+-
+=
rRR
rRdRdP ee
rR = Impedance matchingChoosing R = r to maximize the power delivered to the load resistor.
Experimental Physics II - Electric circuits 10
Combination of resistors
+ -V
I21 VVV += 21 IRIR +=
21 RRR += å= iser RR
21 III += 2211 RIRIV ==
VII
VI
R211 +
==V
RVRV 21 // +=
å -- = 11|| iRR
21
111RRR
+=
1R 2R
+ -V
I
1I
2I
1R
2R
Experimental Physics II - Electric circuits 11
Kirchhoff’s rules
+ -V
1R
2R
3R
+-
When any closed loop is considered, the sum of the changes in electric potential must be zero.
At any branching point the sum of all currents (incoming and outcoming) must be zero.
VVa 0= V331 -=×-= ad VV
0222111 =----- IRIrIRIr ee
2121
21
rrRRI
+++-
=ee
+ -
W= 31R
+-
W= 32Rab
c
dW= 5.0r
e1 = 5 V
e2 = 12 V
V5.85.0112 =×-+= dc VV
V5.531 =×-= cb VVV05.015 =×--= ba VV
A1-=
Experimental Physics II - Electric circuits 12
+ -+
-
Kirchhoff’s rules
W= 31R
W= 32R
12 V
5 V
ab
c
dW= 63R
+ -
+-
ab
c
d
1I
2I 3I
0321 =-+ III
0V12 3211 =+-- RIRI
02A4 21 =+-- IIW3/
0V52332 =+-- RIRI
03A56 32 =-+- II
A1526
1 -=I A1517
2 =I A159
3 -=I
2232RIPR = W08.13
22581
=×= J4.52
=D= tPWR
W71.73=RP W01.9
1=RP W8.17=å RP
W8.2012112=×= ee IP W3535
=×= ee IP
Experimental Physics II - Electric circuits 13
Real life story
3I
321 III +=
e r
xL
r1I
2I
021 =-- rIIxre
0)( 32 =-- IxLrI r
( ) 2IrxLrxx ú
û
ùêë
é+
-+=
rrre
( )rLxLxrI
--=
rrre
23 2L
x =
Experimental Physics II - Electric circuits 14
Simple circuits
+ -
+-
+-
11 re22 re
33 re V
I
0332211 =-+-+- IrIrIr eee
W=== 1321 rrrV111 =e V532 ==ee
A7321
321 =++++
=rrr
I eee
V4111 -=+-=D IrV e
V2222 =+-=D IrV e V2333 =+-=D IrV e
rK ×=eåå+-=+-=D
i
iiiiii r
rKrIrV ee 0=
Experimental Physics II - Electric circuits 15
Adding capacitor – steady state
+ -+
-
W= 31R
W= 32R
5 V
ab
c
dW= 63R
+-3 V
?-cI
?-QPolarity -?
+ -
Adding an open circuit does not affect currents in other parts of the circuit.
0V5V3 =--D+ cV
V8=D cV
µC8V8µF1 =×=D= cVCQ
0V5V3 =-+D- cV
µC2=Q
Experimental Physics II - Electric circuits 16
Charging capacitor
+-I
R
Ce 0=-- IRCqe
0=- IRe:0=t
:¥=t 0=-CQe
0=-- Rdtdq
Cqe ÷
ø
öçè
æþýü
îíì--=RCtCtq exp1)( e
þýü
îíì-=RCt
RtI exp)(
e
Discharge:þýü
îíì-=RCtCtq exp)( e
þýü
îíì--=RCt
RtI exp)(
e
Experimental Physics II - Electric circuits 17
To remember!
Ø Electric current is charge passing through a given area per unit time.
Ø Current density in a conductor is proportional to the electric field.
Ø If a potential difference is maintained across a resistor, the power supplied to it will be equal to the product of the potential difference and the current.
Ø Real battery can be considered as an ideal one connected in series with a small resistance called internal resistance.
Ø For serial connection the resistance are summed; for parallel - their reciprocal values.
Ø When a closed-circuit loop is traversed, the sumin the changes of potential must be zero.
Ø At any junction the sum of inflowing current mustbe equal to the sum of currents flowing out.
Experimental Physics II - Electric circuits 18
To remember!
Ø When a closed-circuit loop is traversed, the sum in the changes of the electric potential must be zero.
Ø At any junction the sum of the inflowing currents must be equal to the sum of the currents flowing out.
Ø No current flows through an open circuit, e.g. through capacitor.
Ø In non-steady state, however, transient currents can exist, e.g. during charging and discharging capacitors.
Ø The product RC is called the time constant of acircuit. It defining, e.g., rate of exponential lost of the charge by capacitor during its discharge.