˙ DALGA TEORIS
˙ I˙
ELEKTROMAGNETIK
˙
˙
KULLANIS¸LI BILG
ILER
~aR · ~ax = sin θ cos φ
~aR · ~ay = sin θ sin φ
~aR · ~az = cos θ
~aθ · ~ax = cos θ cos φ
~aθ · ~ay = cos θ sin φ
~aθ · ~az = − sin θ
~aφ · ~ax = − sin φ
~aφ · ~ay = cos φ
~aφ · ~az = 0
~ar · ~ax = cos φ
~ar · ~ay = sin φ
~ar · ~az = 0
~aφ · ~ax = − sin φ
~aφ · ~ay = cos φ
~aφ · ~az = 0
~az · ~ax = 0
~az · ~ay = 0
~az · ~az = 1
z = R cos θ
p
x2 + y 2 + z 2
θ = cos−1 z/R
φ = tan−1 y/x
x = r cos φ
r =
y = r sin φ
φ = tan−1 y/x
z = z
z = z
x = R sin θ cos φ
y = R sin θ sin φ
R =
~ = ~ax dx + ~ay dy + ~az dz
d`
~ = ~ar dr + ~aφ r dφ + ~az dz
d`
~ = ~aR dR + ~aθ R dθ + ~aφ R sin θ dφ
d`
~ = ~ax dy dz
dS
~ = ~ar r dφ dz
dS
~ = ~aR R2 sin θ dθ dφ
dS
Z
V
~ r) dV =
∇ · A(~
I
dV
= dx dy dz
dV
= r dr dφ dz
dV
= R2 sin θ dR dθ dφ
~ay dx dz
~az dx dy
~aφ dr dz
~az r dr dφ
~aθ R sin θ dR dφ
~aφ R dR dθ
Z
~ r ) · dS
~
A(~
S
~ = ∇Φ + ∇ × F~
A
p
x2 + y 2
~ r) · dS
~=
∇ × A(~
I
S
∇ × ∇Φ ≡ 0
∇ ΦΨ = Φ ∇Ψ + Ψ ∇Φ
~ = Φ ∇×A
~ + ∇Φ × A
~
∇ × ΦA
~ r) · d~`
A(~
C
~ ≡ 0
∇·∇×A
~ = ∇ ∇·A
~ − ∇2 A
~
∇×∇×A
~ = Φ ∇·A
~ + ∇Φ · A
~
∇ · ΦA
∇V
= ~ax
∂V
∂V
∂V
+ ~ay
+ ~az
∂x
∂y
∂z
∇V
= ~ar
1 ∂V
∂V
∂V
+ ~aφ
+ ~az
∂r
r ∂φ
∂z
∇V
= ~aR
∂V
1 ∂V
1
∂V
+ ~aθ
+ ~aφ
∂R
R ∂θ
R sin θ ∂φ
~ = ∂Ax + ∂Ay + ∂Az
∇·A
∂x
∂y
∂z
~ = 1 ∂ rAr + 1 ∂Aφ + ∂Az
∇·A
r ∂r
r ∂φ
∂z
~ =
∇·A
~ = ~ax
∇×A
~ = ~ar
∇×A
~ = ~aR
∇×A
1
∂
1
∂Aφ
1 ∂
2
R
A
+
sin
θA
+
R
θ
2
R ∂R
R sin θ ∂θ
R sin θ ∂φ
∂Az ∂Ay
−
∂y
∂z
1 ∂Az ∂Aφ
−
r ∂φ
∂z
1
R sin θ
+ ~ay
∂Ax ∂Az
−
∂z
∂x
+ ~aφ
∂Ar ∂Az
−
∂z
∂r
∂Aθ
∂
sin θAφ −
∂θ
∂φ
+ ~az
∂Ay ∂Ax
−
∂x
∂y
∂Ar
1 ∂
+ ~az
rAφ −
r ∂r
∂φ
1
+ ~aθ
R
∂ 1 ∂AR
−
RAφ
sin θ ∂φ
∂R
∂AR
1 ∂ + ~aφ
Aθ R −
R ∂R
∂θ
∇2 V
=
∂ 2V
∂ 2V
∂ 2V
+
+
∂x2
∂y 2
∂z 2
2
1 ∂
∂V
1 ∂ 2V
∂ 2V
r
+ 2 2 +
=
r ∂r
∂r
r ∂φ
∂z 2
2
1 ∂
1
∂
∂V
1
∂ 2V
2 ∂V
=
R
+
sin
θ
+
R2 ∂R
∂R
R2 sin θ ∂θ
∂θ
R2 sin2 θ ∂φ2
∇V
∇V
R
dx
R
[x2
±
3
a2 ] 2
xdx
R
[x2 ± a2 ]
R
3
2
R
sin2 x dx =
x 1
− sin 2x
2 4
R
cos2 x dx =
x 1
+ sin 2x
2 4
a2
R
loge x dx = x loge x − x
R
x eax dx =
= −√
x
√
x 2 ± a2
R 2π
0
1
x 2 ± a2
R∞
R∞
0
√
a + x 2 + a2 x
R∞
0
R∞
√
xdx
√
=
x 2 ± a2
x 2 ± a2
0
0
R∞
2
R
1
x
xdx
=
tan−1 2
4
4
2
c +x
2c
c
dx
√
2
x x 2 ± a2
R
0
√
x 2 ± a2
= ∓
a2 x 2
R∞
0
R
2 ax
x e dx = e
ax
2
x
2
2
− 2x + 3
2
a
a
(1 + x)m = 1 + m x +
cos x = 1 −
2 −ax2
xe
R 2π
0
cos2 x dx = π
n!
(a > 0)
an+1
1
2a
(a > 0)
1
2a2
(a > 0)
n!
2an+1
r
(a > 0)
π
a
r
(a > 0)
xe
3
dx =
8a2
2n −ax2
1.3.5 . . . (2n − 1)
dx =
2n+1 an
4 −ax2
x e
R∞
0
eax
(ax − 1)
a2
1
dx =
4a
R ∞ e−nx
√ dx =
0
x
x2
x
dx = x − c tan−1
2
2
x +c
c
2
x3 e−ax dx =
2
dx
1
bx
=
tan−1
2
2
2
a +b x
ab
a
R
2
xe−ax dx =
x2n+1 e−ax dx =
R∞
R
sin2 xdx =
xn e−ax dx =
0
√
dx
√
= loge x + x2 ± a2
x 2 ± a2
dx
1
√
= − loge
2
2
a
x x +a
R
R
1
sin x cos x dx =
sin2 x
2
= ±
dx
1
−1 x
tan
=
x 2 + a2
a
a
R
2
cos(x )dx =
r
(a > 0)
1
sin(x )dx =
2
2
m(m − 1) 2
m(m − 1)(m − 2) . . . (m − k + 1) k
x + ... +
x
2!
k!
x2 x 4
+
− ...
2!
4!
sin x = x −
x3 x5
+
− ...
3!
5!
r
π
a
π
n
R∞
0
π
a
r
π
2
(a > 0)
˙
¨
SABITLER
VE FORMULLER
σ
c = 1−j
F/m
ω
√
k = ω µc = β−jα
1 √
β = α ' √ ωµσ
2
√
σ
β ' ω µ ve α '
2
0 =
r
µ
ω
vp =
m/s
β
2π
λ=
m
β
r
ηc =
µ
Ohm
c
˙ I˙ ILETKENLER
˙
˙ ¸ IN
˙
IY
IC
˙ I˙ DIELEKTR
˙
˙
˙ ¸ IN
˙
IY
IKLER
IC
1
× 10−9
36π
F/m
µ0 = 4π × 10−7
c = 3 × 108
m/s
η0 = 120 π Ohm
H/m
˙
˙ (TM) DUZLEMSEL
¨
PARALEL POLARIZASYONA
SAHIP
DALGALAR:
Γk =
Er0
η2 cos θt − η1 cos θi
=
Ei0
η2 cos θt + η1 cos θi
τk =
2η2 cos θi
Et0
=
Ei0
η2 cos θt + η1 cos θi
˙ POLARIZASYONA
˙
˙ (TE) DUZLEMSEL
¨
DIK
SAHIP
DALGALAR:
Γ⊥ =
Er0
η2 cos θi − η1 cos θt
=
Ei0
η2 cos θi + η1 cos θt
τ⊥ =
Et0
2η2 cos θi
=
Ei0
η2 cos θi + η1 cos θt
Download

sinφ - yarbis