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An overview of maxwell's equations, specifically focusing on the electric field and its relationship to charges. Topics include coulomb's law, the electric force between charges, and the calculation of electric fields for various charge distributions. The document also covers the concept of electric potential and its relation to electric fields.
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∂t
∂ B ∇× E =−
∂ t
∇× H = J + ∂^ D
Maxwell’s Equations point form
What is E? Coulomb’s Law
εο=8.854 x 10-12^ F/m permitivity of free space
k =
1 4 πεο
F
Q Q 4 r
1 2 o 2
= πε
F k Q Q e r 1 2 = (^2) Newtons
Electric Field
1 t o 1t
t a 1 E Q
t^4 R
1 o 1t
πε t N C
V = (^) m 1 volt = (^) CJ = NmC ⇒ (^) mV= NC In general:
E = (^4) πε QoR 2 aR
!! aR is the unit vector from the charge to the point where the field is to be measured
R 1t Qt
r t
r 1
a 1t F t
R 1t = r t - r 1
Qt R = r - r`
r
r` E
Origin
(x,y
,z`)
(x,y,z)
Imaginary Unit Test Charge
[ ]
forn charges 4
E(r)
4 '
' '
' 4 '
E(r)
1 0
0 2 3
m
n m m
Qm
o
Q Q
a r r
r r
r r r r
r r r r
∑ − −
=
−
= − −
− −
=
πε
πε πε
0 3 /^2
0
( ')^2 ( ')^2 ( ')^2
[( ') ( ') ( ') ] 4
4 2
^ − + − + −
= − + − + −
−
−
−
=
x x y y z z
Q x x y y z z
'
' '
Q E(r)
a x a y a z
r r
r r r r
πε
πε
For charge elsewhere:
3 0
3 0
3 0
3
r r
r r
r r
r r
r r
r r
r r
r r
πε
ρ
πε
ρ
πε
ρ
πε
dv
ds
dl
v
s
l
Line C/m
Sheet C/m^2
Volume C/m^3
2 2 23 2 2 2 2
2 2 23 2 2
2 2 2 2 1 2 2 L 2 3
(x a ) (z-z'^ )
dx
1 (x a )
dx
( (z-z') )
(z-z') (a )
dx
=
=
−
=−
= −
ρ
ρ ρ
ρ ρ
ρ ρ
ρ ρ
L L L
L
x
x
x a
x
x
dx (a a
x (^2) (a (^22)
=
x^2 x
3 2 2
1 2
1 ) )
and a ρ
dx dz'
1 dz'
dx
let x (z -z')
=
= −
= −
=
∞
−∞
∞
−∞
Z E L^ a a o (^2222) (z-z')^2
(z-z')
(z -z') 4 ρ ρ ρ
ρ π ε
ρ ρ
Finally:
( )
ρ
ρ
π ε ρ
ρ
π ε ρ
ρ
E a
E a a
o
o
2
0
2 4
L
Z
L
=
+
y z
x y z r a a
r a a a ' y ' z '
x y z = +
= + +
Electric Field Due to a Sheet Charge
'^2 ( ')^2 ( ')^2
' ( ') ( ') x y y z z
x y y z z − = + − + −
− = + − + − r r
r r ax ay az
Q = (^) ∫ ρ s d s d s= d y' d z'
( ' ) 4 [( ') ]
y' z'
4 [ ( ') ( ') ]
y' z'[ ( ') ( ') ]
2 2 2 2 2 2 3
2 2 2 2 3
∫ ∫
∫ ∫ ∞ −∞
∞ −∞
∞ −∞
∞ −∞
= + − − +
=
= + − + −
c x z z y y c
sxd d
x y y z z
sd d x y y z z
o
o
πε
ρ
πε
E ρ^ ax^ ay az
r`
d s=dy’dz’
(y,z
)
(x,y,z)
ρs
2 2 23 2 u^2 c^2
1 u (u c )
u
=
∫ (^) c
d du dy '
( ') = −
u = y − y
' a x ( ') ( ')
' 4 ( ')
1 (^2 2) y y (^2) x (^2) z z 2 dz
y y x z z
sx o
∞
−∞
∞
− ∞
− + + −
− −
= (^) ∫ ρ πε
∫
∞
− ∞ + −
= 2 2 ax ( ')
' ( 2 ) 4
1 x z z
sxdz o
ρ πε
∫ (^) + = a
a
v
v a
dv
arctan 2 2 a x
a x
dv dz
v z z
=
=
= −
= −
2 2
'
( ' )
n o