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

qq

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.