Download EEC 130A : Formula Sheet

EEC 130A : Formula Sheet
Up to Midterm 2
Updated: Mar. 7th 2012
1
Electrostatics
1
E=
4π
Force on a point charge q inside a static electric field
F = qE
Z
l0
Electric field produced by an infinite sheet
of charge
Gauss’s law
E = ẑ
I
D · dS = Q
Electric field produced by an infinite line of
charge
Electrostatic fields are conservative
I
E=
E · dl = 0
or
C
q (R − Ri )
E=
4π0 |R − Ri |3
Z
E = −∇V
dV 0
R02
V =
Electric field produced by a surface charge
distribution
1
E=
4π
Z
S0
E · dl
Electric potential due to a point charge
(with infinity chosen as the reference)
ρv
R̂0
V0
P2
V2 − V1 = −
or
P1
Electric field produced by a volume charge
distribution
Z
Dr
ρl
D
= r̂
= r̂
2πr
Electric field - scalar potential relationship
Electric field produced by a point charge q
in free space
1
E=
4π
ρs
2
∇·D=ρ
or
S
∇×E=0
dl0
R02
ρl
R̂0
q
4π0 |R − Ri |
Poisson’s equation
∇2 V = −
ds0
R02
ρs
R̂0
ρ
Constitutive relationship in dielectric materials
D = 0 E + P
Electric field produced by a line charge distribution
1
where P is the polarization.
Gauss’s law for magnetism
I
∇ · B = 0 or
B · dS = 0
P = 0 χe E
S
Electrostatic energy density
Ampere’s law
1
we = E 2
2
I
∇×H=J
H · dl = I
or
C
Boundary conditions
E1t = E2t
Magnetic flux density — magnetic vector
potential relationship
n̂ × (E1 − E2 ) = 0
or
B=∇×A
D1n − D2n = ρs
or
n̂ · (D1 − D2 ) = ρs
Magnetic potential produced by a current
distribution
Ohm’s law
J = σE
µ
A=
4π
Conductivity
σ = ρv µ
Z
V0
J
dV 0
R0
Vector Poisson’s Equation
where µ stands for charge mobility.
∇2 A = −µJ
Joule’s law
Magnetic field intensity produced by an infinitesimally small current element (BiotSavart law)
Z
P =
2
E · J dv
Magnetostatics
dH =
Force on a moving charge q inside a magnetic
field
F = qu × B
I dl × R̂
4π R2
Magnetic field produced by an infinitely long
wire of current in the z-direction
Force on an infinitesimally small current element Idl inside a magnetic field
H = φ̂
I
2πr
Magnetic field produced by a circular loop
of current in the φ-direction
dFm = Idl × B
Torque on a N -turn loop carrying current I
inside a uniform magnetic field
H = ẑ
Ia2
2(a2 + z 2 )3/2
T=m×B
Constitutive relationship in magnetic materials
where m = n̂N IA.
2
4
Constants
B = µ0 H + µ0 M
Free space permittivity
Magnetization
0 = 8.85 × 10−12
M = χm H
Free space permeability
Boundary conditions
B1n = B2n
or
H1t − H2t = Js
or
µ0 = 4π × 10−7
n̂ · (B1 − B2 ) = 0
n̂ × (H1 − H2 ) = Js
Magnetostatic energy density
1
wm = µH 2
2
3
Useful Integrals
√
dx
= ln(x + x2 + c2 )
x 2 + c2
Z
1
x
dx
= tan−1
2
2
x +c
c
c
Z
√
Z
dx
=
2 3/2
F/m
x
1
√
2
c x 2 + c2
x2 + c
Z
√
x dx
√
= x 2 + c2
x 2 + c2
Z
x dx
1
=
ln (x2 + c2 )
x 2 + c2
2
Z
x dx
1
= −√
2
2
3/2
2
(x + c )
x + c2
Z
dx
1
=−
2
(a + bx)
b(a + bx)
3
H/m