Summary Sheet of EM Formula for Electrostatics and Magnetostatics

UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Supp HO # 4 Prof. Steven Errede
Supplemental Handout #4
Summary of Electrostatics & Magnetostatics
c
=
ε μ
2 1
o
o
Electrostatics:
F = qE + qv × B
E
Gauss’ Law:
ext
1 q
E = ˆ r
2
4πε
o
r
Magnetostatics:
1
F = g B− g v×
E
c
m m 2 m ext
B
g
⎛ μo
⎞ g
= ⎜ ⎟
⎝4π
⎠ r
∇
1 1
E = ρTOT ( ρfree ρbound
)
ε
= o
ε
+
i ∇ i B = 0 (always - no magnetic charges/monopoles)
o
1 enclosed
Φ
E
= ∫ Eda
i = Q
S
TOT
ε
o
1 encl. encl.
= ( Qfree
+ Qbound
)
Φ = m ∫ Bda
i = 0 (n.b. closed surface S)
S
ε
o
∇× E = 0 always, ⇒ E =−∇V
∇× B = μoJ TOT
= μoJ free
+ μoJ
bound
B
= ∇× A
Φ encl
= Ad
m ∫ i
C
Linear Media:
D = εE = εoE
+Ρ
H = 1
B 1 B
μ
=
μ
−Μ
= ( 1 + ),
K = ( )
ε ε χ ε ε
Ρ=
ε χ
o e e o
o
e E
∇× D =∇×Ρ
ρ
bound
=−∇Ρ
nˆ
o
μ = μ 1 + χ , K = μ μ
Μ=
χ H
m
i
bound
o m m o
∇× H =−∇×Μ
J
= ∇×Μ
ρ
bound
σ
bound
=Ρ i
interface
bound
interface
∇ i D = ρ free
∇Ρ=−ρ i
bound
∇× H = J
free
Φ = Dda Q encl
D ∫
=
1
encl
i
S
free
B d = ITOT
μ ∫ i ,
C
o
encl
Φ
Ρ
= ∫ Ρ
encl
ida
=−Q
S
bound
∫ Μ i d = I
C
bound
χ
e
ρbound
=− ρ
free
Jbound = χmJ
free
1 + χ
( )
e
∂ρ
free
∂ρ
bound
∂ρTOT
∇ i J free
=− , ∇ i J bound
=− , ∇ i J TOT
=− ∂ t
∂t
∂ t
ρTOT = ρfree + ρbound
JTOT = J
free
+ Jbound
m
=−∇Μ
i
r
m
ˆ
2
K
=Μ
i bound
nˆ
σ ˆ
m
=Μ i n
interface
∇×Μ = J bound
Hd
= I encl
∫ i
C
free ,
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
1

UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Supp HO # 4 Prof. Steven Errede
Electrostatic Boundary Conditions Magnetostatic Boundary Conditions
2 1 interface ( ⎡
⊥ ⊥
⎤
2 1 interface
0)
( ⊥ ⊥
B )
2
= B ⎡
1 interface
B2 − B ⎤
1 interface
= 0
Ε =Ε ⎣Ε −Ε ⎦ =
⎡
⎣D
− D ⎤
⎦ = ⎡
⎣Ρ −Ρ ⎤
⎦
ε ⎡
o ⎣E
2 1 interface 2 1 interface
⎛∂V
εo⎜
⎝ ∂n
− E ⎤
⎦ = σ
⊥ ⊥
2 1 interface TOT interface
⎣ ⎦
⎡
⎣ − ⎤
⎦ =−⎡ ⎣ − ⎤
⎦ =−
o
= ( σ
free
+ σbound
)
interface
( free bound )
∂V
⎞
− ⎟=−σ
∂n
⎠
2 1
interface interface
TOT
⊥ ⊥ ⊥ ⊥
pole
H2 H1 interface
M2 M1 interface
σ m
interface
1
B B K n
μ ⎡ ⎣ ⎦
1 ⎛∂A
⎜
μo
⎝ ∂n
− ⎤ =
× ˆ
2 1 interface TOT interface
= K + K × nˆ
2 1
interface interface
interface
∂A
⎞
− ⎟=−K
∂n
⎠
=− ( σ
free
+ σbound
) interface
=− ( K
free
+ Kbound
)
TOT
For linear dielectric media:
ε ∂V2 V1
n
ε ∂
− =−
∂
∂n
σ
2 interface 1 interface free
For linear magnetic media:
1 ∂A2 1 ∂A
1
interface
−
interface
=−K
μ ∂n
μ ∂n
2 1
free
⊥ ⊥
⎡
⎣Ρ2 −Ρ ⎤
1 ⎦ interface
=−σ bound
⎡
interface
⎣M ˆ
2
− M ⎤
1 ⎦ interface
= Kbound
× n
interface
⊥ ⊥
⎣
⎡D2 − D ⎤
1 ⎦ interface
= σ free
⎡
interface
⎣H ˆ
2
− H ⎤
1 ⎦ interface
= K
free
× n
interface
2
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.