Chapter 1 Big recap
Dat product ftp.IAIIB cos0 scalar
Cross product written asdeterminant A xD ftp.t É vector
Length Htt A A thxBl is Area ofparallelogram
Triple product Scalar Ä BIE II B E 1 is Volume parallelepiped
see blz 7
Vector A Ex E BIE E Eta B
nevernecessary morethan 1 crossproduct
positionvector
Unitvector I In
Infinitesimaldisplacementvector d dxt dyy.dz E
Seperation vector E F F Z Ik El
sourcepoint
of
position interest
AT FÉdqi Einstein summation Convention used
dt 7T dè
gradiënt vector
Gradiënt 7T Points in direction of maximalincrease offunction T
Magnitude 17hgives slope rateof
increase alongthismaximaldirection
If 7T 0 dt 0 stationarypoint
Del p IE 1
IE Ez E
Divergence p I 3 13 JE
Curl txt
1 En
Vz
Productrules same as forderir À 15 À 7 5 and Nhg Iig
Get scalar fg or I B
Get vector ff or À xD
Page 21 6product rules probalso onformulasheet
, Second derivatives 7 1717
07
Txt 0
Laplacian IT
Integralcalculus Line t.de g a b t.de
jaa surface I dè closedsurface t.de
volume Tdi
É dt dxdydz.frCartesian
de
dr
draaide
sinaardoodfor
for Cylindrical
Spherical
Some important theorems to know
Gradient theorem Htt T b Tca
path independent
de 0
Divergence theorem 7 Ddr I
faucetsin
dat t
uol.me
flowthroughsurface
Strokes theorem t.de t.de
Is ftp.da dependsonlyon boundaryLine notontheparticularsurface used
II dat 0
foranyclosed surface
Integration by parts f g de fgk f.bg
f dx
Cartesian dt dxxntdyy.dzE dt
dxdydz.us
Cylindrical dt ds 5 soldt dzd de
sdsdddz.us
Spherical dt drr rd00 rsinoddodt tsinodrdo.cl0
,Deltafunction ID SU g and Skelet
fix SH flash
3D 8h SexSly Sk and Mde 1
spa
ffHS't âde p
g 4 8127
Helmholtz theorem É if
unambiguously defined 7 E HE specified
and boundary conditions given no r 0
conventioneel minus
If Ix F Ö F TV Fis F or B
Theoremt Curtlessirrotational field
px F Ö everywhere
F de indep ofpath
6 Fde 0 foranyclosedloop
F is gradient of some scalarfunction F TV
If 7E 0 F Pidsome vectorpotential
Theorem 2 Divergenceless solenoidfields
7 F D everywhere
Strada indep ofsurface
EDE 0 foranyclosedsurface
F is the cutofsomevectorfunction F Txt
Always F TV Txt
, Chapter 2
Coulomb's law F I
Electric field E t.IE z
Q is the testcharge
9 are the source charges
Continuous Charge Distribution E Ii f Etr Idq
Along luie dgn IN
i gg 59 Elites Ide
f 9 ij
the prime denotes the source charges Q
p
Field lines density indicates strength density Énn
i
point charge or as vector
If q has 8 than 2g has16
From to
Don't terminate midair but can ago to a
NO INTERSECTIONS
t to source
Flux of É through surface S Eet f E dè fieldlines passing 5
dat product
Gauss's law E dat Itota enclosed charge
het's turn in different.at iso integral
r Edat E de
Using Que Ddt
ME
v e L
t.E e.IT E strijde
o.tt Sir
Integralformbyfareasiestway to compute È if there's symmet
, Example Gaussian surface sphere cylinderor plane
seks
Ë
te
E dij Oey Qen Pdi fles Sds d 2 Kl Sids
kls
We have
IE HETda LEIIda LEI zal
IE tzsl E.ES kls3
E ts
Look at examples 2.5 and 2.6 in the book
strokestheorem
E de 0 y Ê 0
hold for any stal charge distribution
Potential
Define F È dt electricpotential 0 is referencepoint
potentialdifference Kb à È.at Tv
È i convention V of positivecharge positive
Potential obeys superposition V VrtVrt
v NE Volt
Potential not per se zero if E o at tatplace f
Poisson's equation IV
Laplace's equation TV 0 for 8 0
Remember positivecharge potentialhills
negativecharge potentialvalleys DÈ
, VK ffdag
Uci de
This tells how to compute V from given to or 8
Look at example 2.8
For given 8 and symmetry it'smostconvenient tofirst calculate thepotential
P
Triangle En
V
S
E N Én E
SE
Atboundary the normalcomponent of E is discontinuousby
E
Normal derivative In OV in
Work and Energy
F DE W Fd QIE.at Q VIA Na
Forceinapp dir
Path indep Conservative
Hu example
KB Via t
Work neededtoassemble configuration
I work of pointcharges W t TÉ4VII
yougetbackwhendismantled
t potentialenergystored
L IPVde W Gft Vdl W InGouda
W Effende alt space
doesn't takeintoaccountthework necessary tomadetheporiecharge
k more complete totalenergy stored
Use for point charges only
Where is energy stored 41 In field TE energyperunitvolume
2 In Charge IPV energyperunitvolume
1 a NO Superposition I Crossterms Wij WetW 1 EoSÉ Ècht
e.g Doublecharge quadruple totalenergy
Dat product ftp.IAIIB cos0 scalar
Cross product written asdeterminant A xD ftp.t É vector
Length Htt A A thxBl is Area ofparallelogram
Triple product Scalar Ä BIE II B E 1 is Volume parallelepiped
see blz 7
Vector A Ex E BIE E Eta B
nevernecessary morethan 1 crossproduct
positionvector
Unitvector I In
Infinitesimaldisplacementvector d dxt dyy.dz E
Seperation vector E F F Z Ik El
sourcepoint
of
position interest
AT FÉdqi Einstein summation Convention used
dt 7T dè
gradiënt vector
Gradiënt 7T Points in direction of maximalincrease offunction T
Magnitude 17hgives slope rateof
increase alongthismaximaldirection
If 7T 0 dt 0 stationarypoint
Del p IE 1
IE Ez E
Divergence p I 3 13 JE
Curl txt
1 En
Vz
Productrules same as forderir À 15 À 7 5 and Nhg Iig
Get scalar fg or I B
Get vector ff or À xD
Page 21 6product rules probalso onformulasheet
, Second derivatives 7 1717
07
Txt 0
Laplacian IT
Integralcalculus Line t.de g a b t.de
jaa surface I dè closedsurface t.de
volume Tdi
É dt dxdydz.frCartesian
de
dr
draaide
sinaardoodfor
for Cylindrical
Spherical
Some important theorems to know
Gradient theorem Htt T b Tca
path independent
de 0
Divergence theorem 7 Ddr I
faucetsin
dat t
uol.me
flowthroughsurface
Strokes theorem t.de t.de
Is ftp.da dependsonlyon boundaryLine notontheparticularsurface used
II dat 0
foranyclosed surface
Integration by parts f g de fgk f.bg
f dx
Cartesian dt dxxntdyy.dzE dt
dxdydz.us
Cylindrical dt ds 5 soldt dzd de
sdsdddz.us
Spherical dt drr rd00 rsinoddodt tsinodrdo.cl0
,Deltafunction ID SU g and Skelet
fix SH flash
3D 8h SexSly Sk and Mde 1
spa
ffHS't âde p
g 4 8127
Helmholtz theorem É if
unambiguously defined 7 E HE specified
and boundary conditions given no r 0
conventioneel minus
If Ix F Ö F TV Fis F or B
Theoremt Curtlessirrotational field
px F Ö everywhere
F de indep ofpath
6 Fde 0 foranyclosedloop
F is gradient of some scalarfunction F TV
If 7E 0 F Pidsome vectorpotential
Theorem 2 Divergenceless solenoidfields
7 F D everywhere
Strada indep ofsurface
EDE 0 foranyclosedsurface
F is the cutofsomevectorfunction F Txt
Always F TV Txt
, Chapter 2
Coulomb's law F I
Electric field E t.IE z
Q is the testcharge
9 are the source charges
Continuous Charge Distribution E Ii f Etr Idq
Along luie dgn IN
i gg 59 Elites Ide
f 9 ij
the prime denotes the source charges Q
p
Field lines density indicates strength density Énn
i
point charge or as vector
If q has 8 than 2g has16
From to
Don't terminate midair but can ago to a
NO INTERSECTIONS
t to source
Flux of É through surface S Eet f E dè fieldlines passing 5
dat product
Gauss's law E dat Itota enclosed charge
het's turn in different.at iso integral
r Edat E de
Using Que Ddt
ME
v e L
t.E e.IT E strijde
o.tt Sir
Integralformbyfareasiestway to compute È if there's symmet
, Example Gaussian surface sphere cylinderor plane
seks
Ë
te
E dij Oey Qen Pdi fles Sds d 2 Kl Sids
kls
We have
IE HETda LEIIda LEI zal
IE tzsl E.ES kls3
E ts
Look at examples 2.5 and 2.6 in the book
strokestheorem
E de 0 y Ê 0
hold for any stal charge distribution
Potential
Define F È dt electricpotential 0 is referencepoint
potentialdifference Kb à È.at Tv
È i convention V of positivecharge positive
Potential obeys superposition V VrtVrt
v NE Volt
Potential not per se zero if E o at tatplace f
Poisson's equation IV
Laplace's equation TV 0 for 8 0
Remember positivecharge potentialhills
negativecharge potentialvalleys DÈ
, VK ffdag
Uci de
This tells how to compute V from given to or 8
Look at example 2.8
For given 8 and symmetry it'smostconvenient tofirst calculate thepotential
P
Triangle En
V
S
E N Én E
SE
Atboundary the normalcomponent of E is discontinuousby
E
Normal derivative In OV in
Work and Energy
F DE W Fd QIE.at Q VIA Na
Forceinapp dir
Path indep Conservative
Hu example
KB Via t
Work neededtoassemble configuration
I work of pointcharges W t TÉ4VII
yougetbackwhendismantled
t potentialenergystored
L IPVde W Gft Vdl W InGouda
W Effende alt space
doesn't takeintoaccountthework necessary tomadetheporiecharge
k more complete totalenergy stored
Use for point charges only
Where is energy stored 41 In field TE energyperunitvolume
2 In Charge IPV energyperunitvolume
1 a NO Superposition I Crossterms Wij WetW 1 EoSÉ Ècht
e.g Doublecharge quadruple totalenergy
