Flux
Flux Electric through small area: r ^ flux ~ E " n!A Definition of electric flux on a surface: r ^ ! E # n"A
surface
surface
r ^ ! E # n"A
r ^ ! E " ndA r r ! E " dA
r dA
r r electric flux on a closed surface = " E ! dA
Gauss's Law
surface
r ^ " E $ n#A =
"q
inside
!0
r ^ ! E # ndA =
"q
inside
$0
Can derive one from another Gauss's law is more universal: works at relativistic speeds
1 Q E= 4!" 0 r 2
Gauss's law: If we know the field distribution on closed surface we can tell what is inside. 1. Knowing E can conclude what is inside 2. Knowing charges inside can conclude what is E
Gauss's Law for Magnetism
Dipoles: Electric field: `+' and `' charges can be separated Magnetic field: no monopoles Suppose magnetic dipole consists of two magnetic monopoles, each producing a magnetic field similar to the electric field. One cannot separate them total magnetic `charge' is zero.
surface
r ^ " E $ n#A =
"q
inside
!0
surface
r ^ # B " n!A = 0
or
r ^ # B " n!A = 0
Gauss's law for magnetism
Ampre's Law
r r " B # dl = 0 ! I inside _ path
All the currents in the universe contribute to B but only the ones inside the path result in nonzero path integral Ampere's law is almost equivalent to the Biot-Savart law: but Ampere's law is relativistically correct
Inside the Path
Ampere's law r r " B # dl = 0 ! I inside _ path 1. 2. 3. 4. Choose the closed path Imagine surface (`soap film') over the path r r Walk counterclockwise around the path adding up B " dl ! Count upward currents as positive, inward going as negative
!I
inside _ path
= I1 " I 2
!I
inside _ path
= I up
= I up ! I down + I up
Ampere's Law: A Long Thick Wire
r r " B # dl = 0 ! I inside _ path
Can B have an out of plane component? Is it always parallel to the path? r r # B " dl = B2!r
B 2!r = 0 I
for thick wire:
0 2 I B= 4! r
(the same as for thin wire)
Would be hard to derive using Biot-Savart law
Ampere's Law: A Solenoid
r r " B # dl = 0 ! I inside _ path
Number of wires: (N/L)d r r What is B ! dl on sides? B outside is very small r r " B ! dl = Bd
0 IN (solenoid) Bd = 0 I ( N / L ) d L Uniform: same B no matter where is the path
B=
Ampere's Law: A Toroid
r r " B # dl = 0 ! I inside _ path
Symmetry: B || path
B 2!r = 0 IN
0 2 NI B= 4! r
Is magnetic field constant across the toroid?
Maxwell's Equations
Three equations: Gauss's law for electricity
r ^ ! E # ndA =
"q
inside
r Gauss's law for magnetism ^ " B ! ndA = 0 r r Ampere's law for magnetism " B # dl = 0 ! I inside _ path
(incomplete)
$0
Is anything missing?
r r `Ampere's law for electricity' E ! dl = 0 "
Maxwell's Equations (incomplete)
r ^ ! E # ndA =
"q
inside
$0
Gauss's law for electricity Gauss's law for magnetism Incomplete version of Faraday's law Ampere's law (Incomplete Ampere-Maxwell law)
r ^ " B ! ndA = 0 r r " E ! dl = 0 r r " B # dl = 0 ! I inside _ path
First two: integrals over a surface Second two: integrals along a path Incomplete: no time dependence
Is Gauss's Law Relativistically Correct?
r ^ ! E # ndA =
"q
inside
$0
Gauss's law for electricity
Problem: If charges inside move, changes in E will be delayed. To avoid retardation problem: find a property of E at (x,y,z,t) relate it to source charges at (x,y,z,t)
Divergence
r ^ ! E # ndA =
"q
inside
$0
Gauss's law for electricity , where is density of charges
r %"V ^ ! E # ndA = $ 0 r ^ % $ E ndA = " #V !0
`divergence':
r div( E ) # lim
scalar!
r ^ $ E " ndA !V
!V %0
r ! div E = "0
( )
Physical Meaning of Divergence
r ^ ! E # ndA =
"q
inside
$0
Electric field directions `diverge' from a point
r ! div E = "0
( )
Electric field directions `anti-diverge' from a point
Electric field flows through
Is Divergence Relativistically Correct?
r ^ ! E # ndA =
"q
inside
$0
Differential form of Gauss's law
r ! div E = "0
( )
Provides local relationship between charge and electric field E and are at the same location at the same time; that avoids the problem of relativistic retardation!
Integral Form of Gauss's Law
Is integral form of Gauss's relativistically correct? Divergence theorem: r relativistically ! r r div E = ^ correct "0 ! div E dV = ! E " ndA
( )
( )
r ! div E dV =
( )
" ! # 0 dV
Used divergence theorem (pure math) Got Gauss's law
r # "dV ^ # E $ ndA = ! 0 r Q ^ E " ndA = inside # !0
Gauss's law is relativistically correct (unlike Coulomb's law)
Divergence as Partial Derivative
r div( E ) # lim r ^ $ E " ndA !V
!V %0
r E2 !y!z " E1!y!z div( E ) = lim !V #0 !x!y!z r E2 " E1 !E div( E ) = lim = !V #0 !x !x
r !E x !E y !E z + + If E has x, y and z components: div( E ) = !x !y !z
r !Ex !Ey !Ez " div E = + + = !x !y !z # 0
( )
differential form of Gauss's law
Electric Potential and Gauss's Law
r !Ex !Ey !Ez " + + = div E = !x !y !z # 0
( )
!V Ex = " , !x
!V Ey = " , !y
!V Ez = " !z
A boundary value problem
$ 2V $ 2V $ 2V " + 2 + 2 =# 2 $x $y $z !0
Laplace's equation:
! 2V ! 2V ! 2V + 2 + 2 =0 2 !x !y !z
(empty space)
Example Problem
In a certain region of space: V (x, y, z) = V0 + a x 2 + y 2 + z 2 What is the charge density in this region? Solution:
(
)
$ 2V $ 2V $ 2V " + 2 + 2 =# 2 $x $y $z !0
!V = 2 !x
2
! 2 ax 2
!x 2 " 6a = # !0
6a "=# !0
( ) = ! ( 2ax ) = 2a
!x
Standard Notation
r ! ! ! ^ Introduce del operator: " = ^ + ^ + k i j !x !y !z r r " #E x #E y #E z " 1) + + = $#E = #x #y #z ! 0 !0
" ! ^ ! ^ ! ^% ^ E ^ E k = !Ex + !Ey + !Ez i+ j+ k ' Ex i+ y j+ z ^ $ !x !y !z & # !x !y !z
(
)
2)
r & 'V ^ 'V ^ 'V E = ($ $ 'x i + 'y j+ 'z % $ 2V $ 2V $ 2V " + 2 + 2 =# 2 $x $y $z !0
^# k! ! "
3)
divergence of a gradient (Laplacian)
r r E = "! V gradient r r $ ! " !V = # %0 $ 2 ! V = "V = # %0
( )
Is Ampere's Law Relativistically Correct?
r r " B # dl = 0 ! I inside _ path
Problem: If currents inside change, changes in B will be delayed. To avoid retardation problem: find a property of B at (x,y,z,t) relate it to current at (x,y,z,t)
Differential Form of Ampere's Law
r r " B # dl = 0 ! I inside _ path
r ^ Current passing through tiny area: J " n!A r r r ^ # B " dl = 0 J " n!A r r r % B ! dl ^ curl B ! n " lim #A$0 #A Differential form r r different of Ampere's law: curl B = 0 J notation
A J n
( )
( )
r r r " ! B = 0 J
Note: it is a vector equation it is relativistically correct
Gauss's Law
r ^ ! E # ndA =
"q
inside
$0
Divergence theorem: r r ^ ! div E dV = ! E " ndA
( )
r ! div E = "0
( )
Differential form of Gauss's law
Provides local relationship between charge and electric field E and are at the same location at the same time; that avoids the problem of relativistic retardation!
Divergence:
r div( E ) # lim
r ^ $ E " ndA !V
!V %0
r !Ex !Ey !Ez " div E = + + = !x !y !z # 0
( )