!"#$!#!!%
Kirchhoff From The Get-Go
Understanding Circuits
&'(%)*'+*%),%-./0,%1*-1%1*,%20(+34(%506,%74.%
80..,(19%-(:%1*,%;44<%506,%74.%=461-/,9%+-(%>,%
,91->6'9*,:%7.4?%-??,1,.%-(:%@461?,1,.%.,-:'(/9%'(%
9'?<6,%9,.',9%-(:%<-.-66,6%+'.+0'19%'(@,93/-1,:%-9%
,-.6A%-9%BC%4.%B!$D%
M414.'4096A%:'N+061D%
%
20?>6,%47%)'.,9%-(:%>'19%O%<',+,9%
%?,1,.9%1*-1%*44P%0<%'(%70((A%)-A9%
%
Q>91.-+1%':,-9%1*-1%9,,?%14%*-@,%.,6,@-(+,%14%1*,%
R0?>6,%4(%1*,%>,(+*14<%
%
84(709'4(%47%6-(/0-/,%-(:%':,-9%7.4?%<.'4.%6,-.('(/%
E*,A%+-(%>,%09,:%14%946@,%;!%-(:%;F%+'.+0'19%
-64(/9':,%1*,%G,.',9HI-.-66,6%5,:0+34(%?,1*4:D%
%E*,%7-?'6'-.'1A%1*09%,91->6'9*,:%)'1*%J'.+**4K%':,-9%
-(:%<.4+,:0.,9%,-9,9%1*,'.%-<<6'+-34(%14%;L%+'.+0'19%
DDD(41%*,6<,:%>A%1,S1>44P%).'1,.9D%
%
T4.%,S-?<6,U%
Circuit Measurements before Circuit Model
G-?,%-??,1,.%.,-:'(/9%
-.40(:%1*,%+'.+0'1%%
⇒%
?4:,6U %+*-./,%X4)Y%
%
%%+0..,(1D%
%
V-.A%W0(1%
!"#$%&'()'*+,&-+&'.'
V-.A%W0(1%
!"#$%&'()'*+,&-+&'.'
Circuit Measurements before Circuit Model
Z*%A,9Y%1*,A%:4[%
%
E*,%64/'+%'(%1*'9%+-<34(%
\'.,9%%47 %:'K,.,(1%%?-1,.'-69%
%:'K,.,(1%%1*'+P(,99,9%
%
'9%>-+P%14%7.4(1U%
Key Ideas
From Measurements to Model
]91->6'9*%J'.+**4K%-9%90??-.A%47%+'.+0'1%>,*-@'40.%
.'/*1%7.4?%1*,%91-.1%
J'.+**4K%^%
!D  20(+34(%506,%74.%80..,(19%
FD  ;44<%506,%74.%=461-/,9%
Q??,1,.%.,-:'(/9%
=461?,1,.%.,-:'(/9%
E*,%?,-('(/%),%>0'6:%4(%1*,%+'.+0'1%?,-90.,?,(19%
91,?9%7.4?%1*,%4>9,.@,:%<-g,.(9%'(%1*,%(0?>,.9%
E*,%<-g,.(%'(%1*,%-??,1,.%.,-:'(/9%⇒%':,-%47%h+0..,(1b%
E*,%<-g,.(%'(%1*,%@461?,1,.%.,-:'(/9%⇒%':,-%47%h@461-/,b%
Z<,.-34(-6%:,_('34(U%`E*'9%'9%-(%-??,1,.%a%*,.,b9%*4)%A40%09,%'1Dc%
`d01%)*-1%:4,9%'1%?,-90.,ec%
`E*'9%'9%-(%.06,.a%*,.,b9%*4)%A40%09,%'1D%f1%?,-90.,9%6,(/1*c%
DDD1*-1b9%,-9AD%\,%P(4)%->401%6,(/1*D%
!%
!"#$!#!!%
Key Ideas
Using an Ammeter
+0..,(1%
@461-/,%
9,.',9%
<-.-66,6%
Z<,.-34(-6%:,_('34(U%`E*'9%'9%-(%-??,1,.%a%*,.,b9%*4)%A40%09,%'1Dc%
+4?<4(,(19%)'1*%1)4%6,-:9%+-(%
>,%+4((,+1,:%'(%4(6A%1)4%)-A9%
84((,+1,:%'(%9,.',9"&'()''*"+4?<4(,(19%%
,(:H14H,(:U%9,.',9%
9':,H>AH9':,U%<-.-66,6%
!D  d.,-P%1*,%+'.+0'1DDD%
f(@,93/-34(%9,i0,(+,U%
FD  _66%'(%1*,%/-<DDD%
9,.',9%+'.+0'1%
<-.-66,6%+'.+0'1%
+0..,(1%
j-??,1,.%.,-:'(/9k%
!"
#"
@461-/,%
j@461?,1,.%.,-:'(/9k%
$"
%"
Expectations of Ammeter Readings
;,-.('(/%'(%G+',(+,%I.4R,+1%!Cm$9%
j-k  5,:%Q%94+P,1%14%>-g,.A%.,:%
j,@,(10-66Ak%
j>k %d6-+P%8Zl%94+P,1%14%
>-g,.A%>6-+P%j,@,(10-66Ak%
%
Ammeter Readings in the Flashlight Circuit
Bulb On
T40.%+0..,(1%?4:,69%+4??4(6A%90//,91,:%>A%910:,(19%%
!D  94?,%+0..,(1%4(%4(,%9':,Y%(4(,%4(%1*,%41*,.%
FD  94?,%4(%4(,%9':,Y%6,99%4(%1*,%41*,.%j9-AY%8\k%
LD  94?,%4(%4(,%9':,Y%9-?,%4(%1*,%41*,.%j8\k%
"D  +6-9*'(/%+0..,(19%j4(,%9':,%8\Y%41*,.%Q8\%k%
Ammeter Readings in the Flashlight Circuit
Bulb On
Ammeter Readings in a Series Circuit
Bulbs On
&%
,%
-%
+%
'%
F%
!"#$!#!!%
Ammeter Readings in a Series Circuit
Bulbs On
&%
Ammeter Readings in a Series Circuit
Bulbs On
&%
,%
-%
,%
-%
+%
+%
'%
'%
Patterns in the Ammeter Readings
A Possible Model
G,.',9%
G,.',9%
Q!%
84?<4(,(1%
QF%
G-?,%-??,1,.%.,-:'(/9%4(%>41*%
9':,9%47%-%+'.+0'1%+4?<4(,(1U%
QF%
G-?,%-??,1,.%.,-:'(/9%4(%>41*%
9':,9%47%-%R0(+34(U%
20(+34(%
Q!%
Q!%^%QF%
Q!%
84?<4(,(1%
QF%
G-?,%-??,1,.%.,-:'(/9%4(%>41*%
9':,9%47%-%+'.+0'1%+4?<4(,(1U%
QF%
G-?,%-??,1,.%.,-:'(/9%4(%>41*%
9':,9%47%-%R0(+34(U%
Q!%^%QF%
20(+34(%
Q!%
Q!%^%QF%
M41*'(/%'9%6491%
Q!%^%QF%
M41*'(/%'9%6491%
\*,.,%*-@,%),%9,,(%>,*-@'40.%
6'P,%1*-1%>,74.,e%
G1,-:A%X4)%
A Possible Model
f7%1*,%X4)%?4:,6%'9%@-6':Y%1*,(%),%)406:%,S<,+1U%
20(+34(%
Q!%
QF%
QL%
E.-N+%
E41-6%+-.9%,(1,.'(/%R0(+34(%<,.%*40.%
Ammeter Readings in a Parallel Circuit
Bulbs On
Q??,1,.%.,-:'(/9%14%-::%0<%
-+.499%-%R0(+34(U%
Q!%^%QF%n%QL%
\*-1%:4%1*,%-??,1,.%.,-:'(/9%1,66%
09e%
-%
&%
,%
.%
'%
+%
^%
%141-6%+-.9%6,-@'(/%R0(+34(%<,.%*40.%
DDD<.4@':'(/%),%*-@,%
`+4(9,.@-34(%47%+-.9c%
L%
!"#$!#!!%
Ammeter Readings in a Parallel Circuit
Bulbs On
Ammeter Readings in a Parallel Circuit
Bulbs On
-%
&%
,%
-%
&%
,%
.%
'%
+%
.%
'%
+%
Ammeter Readings in a Parallel Circuit
Bulbs On
Ammeter Readings in a Parallel Circuit
Bulbs On
-%
&%
,%
-%
&%
,%
.%
'%
+%
.%
'%
+%
Ammeter Readings in a Parallel Circuit
Bulbs On
Patterns in the Ammeter Readings
5'@,.9%
\-1,.%@460?,%,(1,.'(/%R0(+34(%<,.%9,+4(:%
^%
\-1,.%@460?,%6,-@'(/%R0(+34(%<,.%9,+4(:%
-%
&%
,%
.%
'%
+%
DDD<.4@':'(/%),%*-@,%
`+4(9,.@-34(%47%)-1,.c%
"%
!"#$!#!!%
Patterns in the Ammeter Readings
Patterns in the Ammeter Readings
E.-N+%-(-64/AU%
E.-N+%-(:%.'@,.9%-(-64/A%
⇒%\*-1b9%X4)'(/%'(%-%+'.+0'1e%
DDD<.4@':'(/%),%*-@,%
`+4(9,.@-34(%47%+-.9c%
M0?>,.%47%@,*'+6,9%,(1,.'(/%R0(+34(%<,.%*40.%
^%
%M0?>,.%47%@,*'+6,9%6,-@'(/%R0(+34(%<,.%*40.%
Z>@'409%-(9),.U%
]6,+1.'+%+*-./,%
:.A%93+P%
pQ%
l,-90.,%-%7,)%pQ%
⇒%(4%:'K,.,(+,%>,1),,(%
h91-3+b%-(:%h+0..,(1b%,6,+1.'+'1AD%
Patterns in the Ammeter Readings
Using a Voltmeter
80..,(1U%T64)%47%+*-./,%
84((,+1,:%'(%<-.-66,6%-,/011"+4?<4(,(19%%
20(+34(%506,%74.%80..,(19U%
DDD<.4@':'(/%),%*-@,%
`+4(9,.@-34(%47%+*-./,c%
Voltmeter Readings in the Flashlight Circuit
Bulb On
&%
FD  5,:%%=%94+P,1%14%XA'(/%6,-:%
Voltmeter Readings in the Flashlight Circuit
Bulb On
,%
&%
+%
-%
!D  d6-+P%8Zl%94+P,1%14%>-g,.A%
>6-+P%j<,.?-(,(16Ak%
,%
%E41-6%+0..,(1%'(%^%141-6%+0..,(1%401%
'%
,%
+%
-%
'%
o%
!"#$!#!!%
Voltmeter Readings in the Flashlight Circuit
Bulb On
&%
Voltmeter Readings in the Flashlight Circuit
Bulb On
,%
&%
,%
+%
-%
+%
-%
'%
Voltmeter Readings in the Flashlight Circuit
Bulb On
Patterns in the Voltmeter Readings
G)'1+*%+649,:Y%>06>%4(%
!Do%=%
&%
'%
!Do%=%
,%
+%
-%
$%=%
'%
746:%1*,%644<%401%X-1U%
@461-/,%
.'9,%
Voltmeter Readings in a Series Circuit
Bulbs On
&%
@461-/,%
:.4<%
Voltmeter Readings in a Series Circuit
Bulbs On
,%
-%
&%
+%
'%
141-6%^%$%-.40(:%1*,%644<%
,%
-%
+%
'%
q%
!"#$!#!!%
Voltmeter Readings in a Series Circuit
Bulbs On
&%
Patterns in the Voltmeter Readings
G)'1+*%+649,:Y%9,.',9%>06>9%4(%
,%
!Do%=%
-%
+%
'%
$Dm
o%=
%
$%=%
746:%1*,%644<%401%X-1U%
@461-/,%
.'9,%
Circuit Rules – from patterns in the readings
G4?,1*'(/%:4,9(b1%+*-(/,%-.40(:%1*,%+'.+0'1%
@461-/,%
:.4<%
@461-/,%
:.4<%
141-6%^%$%-.40(:%1*,%644<%
Using Kirchhoff
Q??,1,.%.,-:'(/9%
20(+34(%506,%74.%80..,(19%
%E41-6%+0..,(1%'(%^%141-6%+0..,(1%401%
G4?,1*'(/%:4,9%+*-(/,%-.40(:%1*,%+'.+0'1%
!$%s"
=461?,1,.%.,-:'(/9%
LL%s"
;44<%506,%74.%=461-/,9%
!t%s"
Q66%1*,%@461-/,%+*-(/,9%-.40(:%1*,%+'.+0'1%-::%0<%14%r,.4%
h;44<b%%
⇒ %74664)%94?,%<-1*%>-+P%14%
%)*,.,%A40%91-.1,:%7.4?D%
Solving a Y12 circuit
Solving a Y12 circuit
!$%s"
FDC%="
LL%s"
!t%s"
E)4%)-A9U%
!2 "3'/4'156-/-77'7"8'+9,:0*"" G% %G,.',9Y%G0?%
fGf %I-.-66,6Y%f(@,.9,%47%G0?%47%f(@,.9,9%
LL%##%!t%u%!F%s%n!$%u%FF%s%
FDC%="
!$%s"
LL%s"
!t%s"
FDC%=%v%FF%s%u%$D!L%Q%'(%>-g,.A%>.-(+*D%
=461-/,%-1%14<%R0(+34(%^%FDC%=%:.4<<,:%>A%!DL%=%-+.499%!$%s%^%!Dq%=%%
m%
!"#$!#!!%
Using Kirchhoff
Solving a Y12 circuit
!$%s"
FDC%="
LL%s"
!t%s"
#2 ";4/,<<0="
!$%s"
FDC%="
LL%s"
!t%s"
8'.+0'1%P(4)6,:/,%⇒%9?-66,91%+0..,(1%'(%LL%s%>.-(+*Y%+-66%'1%!!"
⇒%+0..,(1%'(%!t%s%>.-(+*%'9%(,-.6A%F!""jLL#!t%^%!DtLk%
⇒%+0..,(1%'(%!$%s%>.-(+*%'9%(,-.6A%L!"%%>8?"
@8A%-<<6',:%14%6,w%644<%/'@,9%FDC%a%!$xFDtL!%a%LLx!%^%$%7.4?%)*'+*%!%^%$D$"m%Q%
⇒%+0..,(1%'(%!$%s%>.-(+*%'9%FDtLx$D$"m%^%$D!L"%Q%-9%>,74.,%"%
Why use Kirchhoff to solve a Y12 circuit?
Kirchhoff Rules
Q::%-(%,S1.-%>-g,.A%⇒%LL%O%!t%s%.,9'914.9%(4%64(/,.%'(%<-.-66,6Y%94%GI5%7-'69%
!$%s"
LL%s"
FDC%="
/
!t%s"
!$%s
L%=
84(3(0'1A%47%+4(+,<1%:,@,64<?,(1%B!F%14%B!LD%
"
E*'9%+'.+0'1%+-(%>,%946@,:%>A%>41*%GI5%-(:%J'.+**4KY%>01%4(,%?'(4.%
+*-(/,%'(%B!L%-(:%GI5%7-'69D%
Q66%1*-1b9%(,,:,:%14%946@,%-%+'.+0'1%'9%258%%-(:%;5=%<609%-%P(4)6,:/,%47%1*,%
>,*-@'40.%47%+'.+0'1%+4?<4(,(19Y%
,/Y%.,9'914.9%#%^%$!%
Kirchhoff Rules
0
LL%s
!t%s
1
q%=
G46@'(/%9A91,?%47%6'(,-.%,i0-34(9e%
"
\'(14(%9-A9%09,%-%/.-<*'+9%+-6+06-14.%j9,,%*-(:401k%
!"B"#"C"$"D"%"
B!E!"C"E#"C"!F$"D"$"
B!E!"B$$#"C"E$"D"B$
V.-<*'+9%+-6+06-14.%4(%,i0-34(%%%
T!%9'?061-(,409%
TF%74.%L%0(P(4)(9%
](1,.%L%,i0-34(9%
G46@,%
Kirchhoff
Q::%-(%,S1.-%>-g,.A%⇒%LL%O%!t%s%.,9'914.9%(4%64(/,.%'(%<-.-66,6Y%94%GI5%7-'69%
/
L%=
!$%s
0
LL%s
!t%s
1
q%=
B40%+-(%-694%09,%?-1.'S%'(@,.9'4(%4(%]S+,6%
t%
!"#$!#!!%
Kirchhoff Rules
!"B"#"C"$"D"%"
B!E!"C"E#"C"!F$"D"$"
B!E!"B$$#"C"E$"D"B$
/
L%=
!$%s
0
LL%s
V.-<*'+9%+-6+06-14.%4(%,i0-34(%%%
T!%9'?061-(,409%
TF%74.%L%0(P(4)(9%
](1,.%L%,i0-34(9%
G46@,%
!t%s
1
Voltmeter Readings in a Parallel Circuit
Bulbs On
q%=
Voltmeter Readings in a Parallel Circuit
Bulbs On
-%
&%
,%
.%
'%
+%
Voltmeter Readings in a Parallel Circuit
Bulbs On
-%
&%
,%
-%
&%
,%
.%
'%
+%
.%
'%
+%
Voltmeter Readings in a Parallel Circuit
Bulbs On
What is Voltage?
dA%-(-64/A%)'1*%/.-@%I]U%
V.-@%I]%.,6-1,9%14%/.-@%'(1,.-+34(%>,1),,(%?-99,9Y%P/%P'64/.-?9D%
]6,+1.'+%I]%.,6-1,9%14%,6,+1.'+%'(1,.-+34(%>,1),,(%+*-./,9Y%8%+4064?>9D%
-%
&%
,%
.%
'%
+%
C%
!"#$!#!!%
What is Voltage?
What is Voltage?
\-6P'(/%'(%1*,%*'669U%
%
%L$$%2P/H!%
%
14<%<-1*%
L$%?%
" = $%&
! #
>4g4?%<-1*%
"$%P/%
t$%P/%
$%?%
$%2%
$%2%
!$%?%
"$$$%2%
L$%?%
!F$$$%2%
9,-%6,@,6%
$%?%
14<%<-1*%
L$%?%
:4)(*'66%
<-1*%
!$%?%
E*'9%6->,69%1*,%+4(140.%'(:,<,(:,(16A%
47%)*4%)-6P9%1*,%6'(,D%
G4%+-6+06-1,%1*,%VI]%<,.%P/D%
z('74.?%/.-@%_,6:%!$%MP/H!%
:4)(*'66%
<-1*%
%
%!$$%2P/H!%
%
!$%?%
9,-%6,@,6%
$%?%
%
$%2P/H!%
%
>4g4?%<-1*%
"$%P/%
t$%P/%
$%?%
$%2%
$%2%
t$$$%2%
!$%?%
"$$$%2%
t$$$%2%
F"$$$%2%
L$%?%
!F$$$%2%
F"$$$%2%
y'K,.,(1%>4:',9%*-@,%:'K,.,(1%VI]9%
-1%1*,%9-?,%6,@,6D%
Gravitational Potential Energy%
What is Voltage?
%
%L$$%2P/H!%
%
14<%<-1*%
L$%?%
%
%!$$%2P/H!%
%
!$%?%
9,-%6,@,6%
$%?%
%
$%2P/H!%
%
G*'w'(/%-%?-99Y%7.4?%<4'(1%Q%14%<4'(1%d%
'(%1*,%]-.1*b9%/.-@'1-34(-6%_,6:%'(@46@,9%
-%/.-@'1-34(-6%<41,(3-6%,(,./A%+*-(/,%
47%!$%2P/H!Y%'(%1*'9%+-9,D%
:4)(*'66%
<-1*%
>4g4?%<-1*%
\*-1b9%1*,%%
%:'K,.,(+,%'(%/.-@'1-34(-6%<41,(3-6%,(,./A%<,.%P'64/.-?%
%>,1),,(%1*,%14<%<-1*%-(:%1*,%>4g4?%<-1*e%
d%
1m
]'1*,.%)4.P%'9%:4(,%0*%1*,%_,6:Y%
'(+.,-9'(/%1*,%'(1,.-+34(%,(,./AY%VI]Y%
'7%1*,%?-99%'9%.-'9,:D%
Q%
Z.%)4.P%'9%:4(,%&G%1*,%_,6:Y%
:,+.,-9'(/%1*,%'(1,.-+34(%,(,./AY%VI]Y%
'7%1*,%?-99%'9%64),.,:%4.%:.4<<,:D%
\,%+-(%1-6P%->401%1*,%/.-@'1-34(-6%<41,(3-6%,(,./A%:'K,.,(+,%<,.%P'64/.-?%
>,1),,(%Q%-(:%d%
⇒%/.-@%Iy%^%F$$%2P/H!%
Electric Potential Energy%
Coulomb%
;'P,)'9,Y%9*'w'(/%-(%,6,+1.'+-66A%+*-./,:%<-.3+6,Y%
,/Y%-(%,6,+1.4(Y%7.4?%<4'(1%Q%14%<4'(1%d%'(%-(%
,6,+1.'+%_,6:%'(@46@,9%-(%,6,+1.'+%<41,(3-6%,(,./A%
+*-(/,%47Y%9-A%'(%1*'9%+-9,Y%!Dq%x%!$H!C%2%<,.%
,6,+1.4(D%%
E*-1b9%-%@,.A%9?-66%-?40(1%47%,(,./AD%
Q(:%'(%-(A%+-9,Y%,@,.A:-A%,6,+1.'+-6%
,@,(19%'(@46@,9%*0/,%(0?>,.9%47%
,6,+1.4(9D%
d%
84064?>%0(+4@,.,:%1*,%.,6-34(%>,1),,(%
1*,%7-+14.9%1*-1%:,1,.?'(,%1*,%,6,+1.'+%
'(1,.-+34(%74.+,%>,1),,(%+*-./,:%4>R,+19%
\,%/.40<%,6,?,(1-.A%+*-./,9%
6'P,%,6,+1.4(9%'(14%>0(:6,9%
+-66,:%`+4064?>9c%
1m
Q%
!%+4064?>Y%!%8Y%'9%1*,%141-6%+*-./,%
+-..',:%>A%qDFo%x%!$!t%,6,?,(1-.A%
+*-./,9%
j%,/Y%,6,+1.4(9%4.%<.414(9k%
G'S%-(:%-%i0-.1,.%>'66'4(%>'66'4(%
,6,?,(1-.A%+*-./,9D%
!$%
!"#$!#!!%
Electric Potential Energy%
;'P,)'9,Y%9*'w'(/%-(%,6,+1.'+-66A%+*-./,:%
<-.3+6,Y%,/Y%-(%,6,+1.4(Y%7.4?%<4'(1%Q%14%
<4'(1%d%'(%-(%,6,+1.'+%_,6:%'(@46@,9%-(%
,6,+1.'+%<41,(3-6%,(,./A%+*-(/,%47Y%9-A%
'(%1*'9%+-9,Y%!Dq%x%!$H!C%2%<,.%,6,+1.4(D%%
What is Voltage?
G+-6'(/%0<%14%-%+4064?>U%
E*,%,6,+1.'+%<41,(3-6%,(,./A%:'K,.,(+,%
>,1),,(%Q%-(:%d%'9%%
!Dq%x%!$H!C%2%<,.%,6,+1.4(%
)*'+*%'9%
d%
!Dq%x%!$H!C%2%<,.%,6,+1.4(%x%qDFo%x%!$!t%
,6,+1.4(9%<,.%+4064?>%
%
4.%!%R406,%<,.%+4064?>%%
Q%
E*,%'(91.0?,(1%1*-1%?,-90.,9%
,6,+1.'+%<41,(3-6%,(,./A%:'K,.,(+,%
<,.%+4064?>%>,1),,(%<6-+,9%'9%
+-66,:%-%@461?,1,.%
1m
E*,%@461-/,%j4.%Iy%<41,(3-6%
:'K,.,(+,k%.,<4.19%1*,%R406,9%<,.%
+4064?>%>,1),,(%1*,%1)4%<6-+,%
!%@461%^%!%R406,%<,.%+4064?>%
!%=%^%!%28H!%
Resistance
M4)%-./0,%>A%-(-64/A%14%,6,+1.'+%Iy%
%
%L$$%28H!%
%
z('74.?%,6,+1.'+%_,6:%!$$$%M8H!%
$DL%?%
%
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%
$D!%?%
\*-1b9%1*,%%
%:'K,.,(+,%'(%,6,+1.'+%<41,(3-6%,(,./A%<,.%+4064?>%
%>,1),,(%1*,%14<%<-1*%-(:%1*,%>4g4?%<-1*e%
⇒%,6,+1.'+%Iy%^%F$$%28H!%^%F$$%=%
Resistance
Q%+'.+0'1%+4?<4(,(1Y%
,/Y%1*,%10(/91,(%)'.,%_6-?,(1%'(%1*,%6-?<Y%
%1*-1%1.-(97,.9%,(,./A%7.4?%1*,%,6,+1.'+-6%
914+P%14%1*,%1*,.?-6%,(,./A%914+P%'9%9-':%14%
*-@,%-%`.,9'91-(+,cD%
f1%'9%9-':%14%>,%-%`.,9'914.c%
Q%@461-/,%-<<6',:%-+.499%1*,%6-?<%9,19%0<%-(%,6,+1.'+%_,6:%'(%1*,%)'.,D%%
T.,,%,6,+1.4(9%-++,6,.-1,%'(%1*,%,6,+1.'+%_,6:Y%'(+.,-9'(/%1*,'.%P'(,3+%,(,./A%
)*'6,%1*,%<41,(3-6%,(,./A%914+P%47%1*,%_,6:%:,+.,-9,9D%
Z(%-@,.-/,Y%7-91%7.,,%,6,+1.4(9%+466':,%)'1*%-14?9%'(%1*,%)'.,%-(:%964)%:4)(D%
E*,'.%J]%'9%1.-(97,..,:%14%1*,%'(1,.(-6%,(,./A%47%1*,%)'.,%a%1*,%)'.,%/,19%*41D%
!EE"H"
/'141(0/%
E*'9%?4:,6%6,-:9%14%-%.,6-34(%>,1),,(%1*,%+0..,(1%1*.40/*%1*,%.,9'914.%-(:%
1*,%@461-/,%-+.499%'1%1*-1%)-9%:'9+4@,.,:%,S<,.'?,(1-66A%>A%Z*?%'(%1*,%!tF$9%
Resistance
Resistance
!EE"H"
/'141(0/%
!EE"H"
/'141(0/%
!!%
!"#$!#!!%
Resistance
Resistance
!EE"H"
/'141(0/%
Resistance
=461-/,%-+.499%.,9'914.%#%=%
Resistance
%I"H"
/'141(0/%
964<,%%
^%"%=%#%"$%?Q%
^%!$$%s%
%I"H"
/'141(0/%
80..,(1%1*.40/*%.,9'914.%#%?Q%
Resistance
Definition of Resistance
=461-/,%-+.499%.,9'914.%#%=%
V.-<*9%90//,91%
!$$%s%
z('19%-.,%@4619%<,.%-?<%a%.,(-?,:%4*?%
Q>>.,@'-1,%4*?%14%Z%
964<,%%
^%"Dm%=%#%!$$%?Q%
^%"m%s%
%1*-1b9%(4%/44:%a%%
Q>>.,@'-1,%4*?%14%s%
'9%!$$%Z%-%*0(:.,:%4*?9%4.%-%1*409-(:%4*?9%)'1*%1*,%0('1%6,w%4Ke%
80..,(1%1*.40/*%.,9'914.%#%?Q%
!F%
!"#$!#!!%
Ohm’s Law
Mental Model of a Metal
;-{+,%47%<49'3@,%-14?'+%+4.,9%'??,.9,:%'(%9,-%47%
?4>'6,%,6,+1.4(9%
l4>'6,%,6,+1.4(%
d40(:%,6,+1.4(%
Z*?b9%6-)U%l,1-69%-1%-%+4(91-(1%1,?<,.-10.,%*-@,%-%+4(91-(1%.,9'91-(+,D%
G0+*%?-1,.'-69%-.,%+-66,:%4*?'+%+4(:0+14.9%
l,1-6%,6,+1.'+-66A%(,01.-6%⇒%(4%4@,.-66%+*-./,%9,<-.-34(%
⇒%j!k %.-(:4?%?434(%
%E*,.?-6%?434(%47%?4>'6,%,6,+1.4(9%|%!$o%?9H!%
%
⇒%jFk %'(1,.(-6%,6,+1.'+%_,6:%^%$%
Mental Model of a Metal
Conduction
l,1-6%,6,+1.'+-66A%(,01.-6%
Q%91.-'/*1%)'.,%)'1*%-%91,-:A%+0..,(1%'(%
'1%644P9%6'P,%1*'9%(4)%
n%
d,-1Ab9%:,?4U%
6-{+,%
^%
,6,+1.4(%9,-%
?,1-6%
DDD-(:%6-1,.%
l4>'6,%,6,+1.4(9%:4(b1%<09*%,-+*%41*,.%6'P,%<,-9%'(%<'<,D%
l,1-6%'(%,S1,.(-6%%
,6,+1.'+%_,6:%
E*,.,%-.,%-9%?-(A%'(%7.4(1%<09*'(/%>-+P%-9%>,*'(:%<09*'(/%74.)-.:%
G4%1*,.,%?091%>,%-(%,S1,.(-6%:4664<%47%+*-./,%94?,)*,.,%14%9,1%0<%
-(%,6,+1.'+%_,6:%'(9':,%1*,%)'.,%
I46-.'9,:%a%(41%+*-./,:%
:'<46,%_,6:%
]6,+1.'+%_,6:%'(9':,%
<46-.'9,:%?,1-6%'9%r,.4%
⇒%964)%:.'w%@,64+'1A%-::,:%14%
.-<':%1*,.?-6%?434(%
,S1,.(-6%
DDD1*,%y.0:,%?4:,6%
\*,.,e%
E*,%>-g,.A%1,.?'(-69e%
Conduction
G4%),%?'/*1%,S<,+1U%
G1.4(/%_,6:%
\,-P%_,6:%
d.'/*1%>06>%%
y'?%>06>%
DDD)*'+*%),%:4(b1%9,,D%
G4%)*,.,%-.,%1*,%+*-./,9%1*-1%
?-P,%,6,+1.'+%_,6:9%'(%)'.,9e%
!L%
!"#$!#!!%
Where are the charges?
Where are the charges?
]6,+1.'+%_,6:%7.4?%>-g,.A%+*-./,9U%
>-g,.A%
<6-+,9%47%'(1,.,91%%x%
>,(1%)'.,%
!%%
x%
F%%
x%
!%%
x%
F%%
x%
L%%
x%
L%%
x%
"%%
x%
"%%
x%
o%%
x%
o%%
x%
Where are the charges?
Where are the charges?
5,9063(/%,6,+1.4(%9,-%:.'wU%
84(9,i0,(1%>0'6:0<%47%+*-./,%-1%1*,%>,(:9U%
!%%
x%
F%%
x%
!%%
x%
L%%
x%
F%%
x%
L%%
x%
0<91.,-?%-1%"e%
"%%
x%
"%%
x%
o%%
x%
,/Y%.,:0+'(/%
-(:%,@,(10-66A%
.,@,.9'(/%1*,%
_,6:%-1%"%
o%%
x%
Where are the charges?
Surface Charges?
DDD,@,(10-66A%141-6%_,6:%47%>-g,.A%-(:%<'6,:H0<%+*-./,9%'9%
0('74.?%-(:%:'.,+1,:%-64(/%1*,%)'.,%,@,(%.40(:%1*,%>,(:9%
nnnn%
H%H%H%H%
nnnn%
H%H%H%H%
E*,%<'6,:H0<%
+*-./,%?091%>,%
4(%1*,%90.7-+,%
nnn%
nnn%
H%%%%
n%
H%%%%
n%
H%H%%%
H%H%%%
n%%%%%%n%
n%%%%%%n%
H%H%H%%
H%H%H%%
!"%
!"#$!#!!%
Patterns in the Voltmeter Readings
20(+34(%
G)'1+*%+649,:Y%9,.',9%>06>9%4(%
Q!%
QF%
QL%
$%=%
!Do%=%
$%=%
746:%1*,%644<%401%X-1U%
$%=%
!Do%=%
@461-/,%
.'9,%
@461-/,%
:.4<%
@461-/,%
:.4<%
$Dm
o%=
%
141-6%^%$%-.40(:%1*,%644<%
!o%