1 Termodinamika és optika gyakorlat II. éves fizikushallgatók ...

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¥
¥§
§
¥
¥
¥
¨¤
§
¥ 2
p = p0 − αV
¥
Cx = DQ
dT
DQ = dU + pdV = ∂U
∂T
Cx = ∂U
∂T
V
+
V
∂U
∂V
¥
∂U
p = f(V )T
∂V = 0
T
¥ CV = DQ
dT
V
= ∂U
∂T V
Cx = ∂U
∂T
V
x
=?
dT + ∂U
∂V
T
+ p ∂V
∂T
Cx = CV + p ∂V
∂T
T
∂V
+ p
∂T
x
dV + pdV
p − p0
V 2
p = p0 − αV 2
x
= α = const.
x

¥
pV = nRT
p0 − αV 2 V = nRT
p0V − αV 3 = nRT
p0
∂V
∂T
α
∂V
∂T
2 ∂V
− 3αV
∂T
α
=
nR
∂
∂T
α
p0 − 3αV 2
Cα = CV + p0 − αV 2
α
= nR
nR
p0 − 3αV 2
p/V = const Cx
= Cp+Cv
¥
2
T (V ) =
p0 − αV 2 V = nRT
T (V ∗ ) = max.
p0 − αV 2 V
nR
= p0 − αV 3
nR
dT (V )
dV = 0 = p0 − 3αV 2
nR
V ∗ =
Tmax = T (V ∗ ) = p0
p0
3α
p0
3α
p0 p0
− α 3α 3α
nR
= 2p0
p0
3nR 3α

§
£
£
p + an2
V 2
(V − bn) = nRT
U = cV mT − n2 a
V
0 = DQ = dU + pdV
¥ p = nRT
V −bn
m/n = M ¥
an2 − V 2
0 = DQ = cV mdT + n2
a nRT an2
dV + −
V 2 V − bn V 2
dV
0 = cV mdT + nRT
V − bn dV
cV m nR
dT = −
T V − bn dV
cV ln T + const1 = − R
ln (V − bn) + const2
M
ln T cV + ln (V − bn) R/M = const3
T cV (V − bn) R/M = const

¥
p = 1
3 aT 4 ,
U = aV T 4 .
DQ = dU + pdV = aT 4 dV + 4aV T 3 dT + 1
3 aT 4 dV = 4
3 aT 4 dV + 4aV T 3 dT = 0,
− 1 dV
3 V
= dT
T .
1 − 3 V
ln = ln
V0
T
.
T0
T V 1/3 = const.
pV 4/3 = const.
¥ γ−1 T
V
γ = cp/cV
¨¤
§
¥ §¥
T1
T2
> T1
T1
¥
T1 T2
V = conts
η = W
Qfel
= Qfel − |Qle|
.
Qfel
Q12 = cV m(T2 − T1) > 0, felvesz.
Q23 = 0
¥ ¥
dU = 0 DQ = dU + pdV =
pdV
Q31 =
¥ T1V γ−1
3
p dV = m
M RT1
V3
V1
1
V
= T2V γ−1
1
m
dV =
M RT1 ln V1
< 0, lead.
V3
Q31 = 1 m
γ − 1 M RT1 ln T1
= cV mT1 ln
T2
T1
,
T2

p
1
2
V1
¥
izoterma
T
2
adiabata
cp − cV = R
M .
η(T1, T2) = 1 − |Q31|
cV mT1 ln
= 1 −
T1
T2
cV m(T2 − T1)
Q12
DQ = T dS S
dU = T dS + DW.
CV = T ∂S
∂T
dU = T dS − pdV.
V
Cp = T ∂S
,
∂T
α = 1
V
κT = − 1
V
p
,
∂V
,
∂T
p
∂V
∂p
T
3
V 3
T1
= 1 − T1
T2 − T1
V
ln T2
.
T1
V
p
,

¥
∂ 2 f(x, y)
∂x ∂y
∂f
∂x
z
∂x
∂y
x y z
¥
z
df = ∂f
∂x
y,z
= ∂2 f(x, y)
∂y ∂x ,
= ∂f
∂y
=
z
∂y
∂x
∂y
,
∂x
z
z
−1
.
f(x, y, z) = 0,
dx + ∂f
∂y
∂y
∂x
z
x,z
= −
dy + ∂f
∂z
∂f
∂x
y
,
∂f
∂y
x
x,y
dz = 0
¥
x y z
∂x
∂y
∂y
∂z
∂z
∂x
= −1,
∂x
∂y
z
z
= −
x
∂z
∂y
x
∂z
∂x y
§
§
α κT
¥
§
∂p
∂T
V
,
= −
∂α
∂p
y
¥
¥
∂p
∂T
V
∂V
∂T p
∂V
∂p
T
T
= V α
=?
= − ∂κT
∂T
−V κT
.
p
= α
.
κT

§
−
1 ∂ V
∂V
∂T p
∂p
∂ − 1
V
dU = T dS + pdV
∂T
∂ 1
T
∂V
∂p
T
T
= − 1
V 2
p
∂U
∂V
∂S
∂V
∂S
∂T
∂U
∂V T
∂T
= − 1
V 2
T
∂V
∂p
T
∂V
∂T
= T ∂p
∂T
V
p
∂V
∂T
p
∂V
∂p
− p.
dS = 1 p
dU +
T T dV.
T
V
∂U
∂V
T
= 1
T
= 1
T
∂U
∂V
∂U
∂T
T
V
∂2S ∂T ∂V = ∂2S ∂V ∂T ,
V
− 1
T 2
= T 2
1
T
+ ∂ p
T
∂T
V
+ 1
V
T
+ p
T ,
.
= ∂ 1
T
∂V
∂ 2 U
∂T ∂V = ∂2 U
∂V ∂T .
∂U
∂V
∂p
∂T
T
V
+ ∂ p
T
∂T
V
− p
T 2
= 0.
+ 1
V
∂U
∂T V
= T ∂p
∂T
∂ 2 V
∂p ∂T .
T
∂ 2 V
∂p ∂T .
¥¥
V
,
− p.

¨¤
dU = T dS − pdV.
¥
U(S, V, N)
F (T, V, N)
H(S, p, N)
G(T, p, N)
§
dU = T dS − pdV + µdN
dF = −SdT − pdV + µdN
dH = T dS + V dp + µdN
dG = −SdT + V dp + µdN
∂U
∂V
T
= T ∂p
∂T
§
¥
∂U
∂V
T
V
dU = T dS − pdV,
= T ∂S
∂V
∂S
∂V
∂T
∂V
T
T
∂T
∂V
S
− p.
− p = T ∂p
∂T
= ∂p
∂T
S
= −
=?
V
.
∂S
∂V T
∂S
∂T V
−SdT − pdV
∂p
∂T ∂T
V T α
= − = − .
∂V CV /T CV κT
S
.
V
− p.

§
∂T
∂p
¥ ¥
−SdT + V
dp
§
∂U
∂p
T
= T ∂S
∂p
T
∂T
∂p
S
= −
− p ∂V
∂p
T
∂S
∂p
T
∂S
∂T p
∂U
∂p
S
=
T
=?
∂V
∂T p
Cp/T
=?
= −T ∂V
∂T
−SdT + V dp
§
T0
∂V
∂T
p
V = a1 − a2p + a3p 2 ,
p
= a4 + a5p,
a1, a2, a3, a4, ¥
a5
W ∆U
T V α
= .
Cp
+ pV κT = −T V α + pV κT .
T0
DW = −pdV = −p(−a2dp + 2a3pdp)
p2
W = −
p1
−a2p + 2a3p 2 dp = a2
∆U = Q + W T0
Q =
p2
p2
T0dS = T0
∂S
∂p
dp = −T0
p1
= −T0
p1
a4 (p2 − p1) + a5
∆U = p2 1 − p 2 2
2
T
p2 1 − p2
2
.
2
p2
p 2 2 − p 2 1
2
p1
∂V
∂T
p
− 2
3 a3
3
p2 − p 3 1 .
dp = −T0
p2
(a2 − a5T0) − 2
3 a3
3
p2 − p 3 1 − a4T0 (p2 − p1) .
p1
p1 p2
(a4 + a5p) dp

§
∂ T ∂S
∂T p
∂p
T
= T ∂2 S
∂p∂T
= −T ∂V
∂T
∂Cp
∂p
T
= T
p
= −T V α 2 + ∂α
.
∂T
∂
∂S
∂p
T
∂T
α − T V ∂α
∂T
p
p
¨¤
u(x, y) v(x, y)
∂ (u, v)
∂ (x, y) =
§
§
−V κS = ∂V
∂p
= ∂S
∂T
∂u
∂x y
∂u
∂y
x
∂ (u, z)
∂ (x, z)
∂ (u, z)
∂ (z, x)
∂ (u, v)
∂ (x, y)
S
V
∂v
∂x y
∂v
∂y
x
∂u
=
∂x
∂u
= ,
∂x
z
y
∂u
= − ,
∂x
z
= ∂ (u, v)
∂ (p, q)
p
∂V ∂
∂T p
= −T
∂T
p
p
= −T
= −T V α 2 + ∂α
.
∂T
∂v
∂y
κS = CV
κT .
Cp
x
∂ (p, q)
∂ (x, y) .
− ∂u
∂y
x
∂v
.
∂x
= ∂ (V, S) ∂ (V, S) ∂ (V, T ) ∂ (p, T )
=
∂ (p, S) ∂ (V, T ) ∂ (p, T ) ∂ (p, S) =
∂V
∂p
1
∂S = Cv
T (−V κT ) T
Cp
= −V CV
Cp
T
∂T p
T V α2
Cp − CV = .
κT
y
κT .
∂ (V α)
∂T
p

Cp
T
∂ (S, p) ∂ (S, p) ∂ (T, V )
= =
∂ (T, p) ∂ (T, V ) ∂ (T, p) =
= CV
T
= CV
T −
= CV
− ∂S
∂V
¥
§
T
∂p
∂T
V
∂p
T
∂p
∂T
V
2 ∂V
∂p
T −
⎛ ⎞
∂V −
∂T ⎝ p
⎠
∂V
2
∂V
∂p
T
∂V
∂p
T
T
∂S
∂T
V
∂p
∂V
T
− ∂S
∂V
T
∂p
∂T
V
− SdT − pdV )
¥
= CV
T −
∂V
∂T
p
∂V
∂p
T
2
= CV
T − V 2α2 .
−V κT
∂V
∂p
S = aT a = a(p, V ) T
Cp − CV ∼ T 3 .
£ 2
T α ¥
κT
S
α = 1
V
κT = − 1
V
∂V
∂T
p
∂V
∂p
= − 1
V
T
= 1
V
∂S
∂p
T
∂V
∂S
T
∼ T,
∂S
∂p
¥
1/T T κT ∼ 1
2 T T
Cp − CV ∼
1 = T 3 .
§
CV
T
= 1
V
¥
T
p = A(V )T + B(V )
∂S
∂V
1
T
∂S
∂p
¥ V
0 =
∂S ∂
∂T V
∂V
T
= ∂ ∂S
∂V T
∂T
V
=
∂ ∂p
∂T
V
∂T
V
= ∂2p .
∂T 2
p T p = A(V )T + B(V )
§
∂V
∂T S
∂V
∂T
p
= 1
1 − γ ,
γ = Cp/CV .
T
.
T

∂V
∂T S
∂V
∂T
p
=
− ∂S
∂T | V
∂S
∂V | T
V α
= − CV
T V α
= − CV
T V α
1
− ∂V
∂T p
∂S
∂V
∂V
∂p
1
T
T
= − CV
T V α
= CV
T V α
¥
− CV κT CV
= −
T V α2 Cp − CV
¨¤
1
∂p
∂T
V
1
V α (−V κT ) = − CV κT
.
T V α2 = 1
1 − γ .
¥
M = f(H, T ),
dU = T dS + µ0HdM,
f ££
H M
M H M H £
¥
¥
M ←→ V,
µ0H ←→ −p.
§
CH
T
∂ (S, H) ∂ (S, H)
= =
∂ (T, H) ∂ (T, M)
= CM
T
+ µ0
∂H
∂T
M
CH − CM = µ0T χT V
χT = 1
V
∂ (T, M)
∂ (T, H) =
2 ∂M
∂H
T
,
∂S
∂M
∂H
∂M
∂H
∂S
∂T
= µ0
T
∂T M
∂H
∂T
T
M
∂H
∂T
.
M
M
∂H
∂M
2
,
T
− ∂S
∂M
T
∂H
∂T
M
∂M
∂H
¥ H χT > 0 χT < 0
T

§
¥
M =
C CH − CM
χT = 1
V
CH − CM = µ0T C
T
∂H
∂T
M
CV H
T ,
∂M
∂H
T
= M
CV .
= C
T .
2 M
V
C2 M
= µ0
V 2 2
CV
§
M = V f
H
T
¥
U M T
dU = T dS + µ0HdM
∂U
= T
∂M
∂S
∂M
T
T
+ µ0H = −µ0
∂ (H/T )
∂T
M
= 1
T
∂H
∂T
∂H
∂T
M
M
= µ0
= −µ0T
− H
T 2
CV H2 T 2 .
2 ∂ (H/T )
∂T
M = const. f(H/T ) = const H/T = const.
∂U
∂M
§
dH
¥
DQ = T dS = 0
0 = dS = ∂S
∂T
¥
H
dT = − µ0T
CH
T
dT + ∂S
∂H
∂M
∂T
H
T
= 0.
dH = CH
T
dH = µ0CV H
CH
¥ CV H M = T
H T
+ µ0
∂M
∂T
1
T dH.
H
.
M
.

§
¥ ¥
H → 0 C0 = B/T 2 B
¥
T1
H1
H2 < H1
¥
U M
CM =
∂U (M, T )
∂T
H
M
CH = CM + µ0CV H 2
¥
dT = µ0CV H
CH
T 2
= ∂U (T )
1
dH = T
T
∂T
= C0.
= B + µ0CV H2 T 2 .
µ0CV H
dH.
B + µ0CV H2
T (H)
dT
T =
H
dH.
B/ (µ0CV ) + H2
T2 < T1
x
a 2 +x 2 = 1
2 ln a 2 + x 2
T2 = T1
B + µ0CV H2 2
B + µ0CV H2 .
1
H2
< H1 B ≪ µ0CV H2 2
H1
T2 ≈ T1 H2

¨¤
§
¥§
¥
r £ d
n > 1
θ
r
§
sin θ = r
r+d
r 1
>
r + d n , r > d
n − 1
d
sin θ > 1/n
§
z n(z)
z θ(z)
§
¥
z0
θ(z0)
n ∝ √ ρ ρ
§
z0
z0 +∆z §
∆θ z0 z0+∆z0
R(z0) = | lim∆z→0(∆l/∆θ)| ∆l z0
∆l
R(z0) = lim
∆z
∆l 1
∆z→0 ∆z ∆θ = lim
∆z→0 ∆z |θ ′ (z0)| .
lim
∆z→0
∆l
=
∆z
1
cos[θ(z0)] .

z
z0 + ∆z
z0
∆θ
∆l
z
z0 + ∆z
z0
z0
θ(z0 + ∆z)
θ(z0)
z0 + ∆z
§£
n(z0) n(z0 + ∆z)
sin(θ(z0))
sin[θ(z0 + ∆z)] = n(z0 + ∆z)
n(z0)
1
sin[θ(z0 + ∆z)] ≈
1
sin[θ(z0)] − ∆z cos[θ(z0)]θ ′ (z0)
sin 2 [θ(z0)]
¥¥ ¥
z0
1 − ∆zctg[θ(z0)]θ ′ (z0) = 1 + ∆z n′ (z0)
n(z0) .
z0
1
θ ′ n(z0)
= −
(z0) n ′ (z0) ctg[θ(z0)]
R(z0) =
1
|[ln ◦n] ′ (z0) sin[θ(z0)]|
ρ(z) = ρ0 exp − Mg
T
M
g
R0
z n(z) = n0 exp − Mg
2R0T z
− Mg
2R0T
R(z0) = 2R0T
Mg
R0T z
(ln ◦n)(z) = ln(n0) − Mg
2R0T z (ln ◦n) ′ (z) =
1
|sin[θ(z0)]|
§
§
n = 4/3
¥
¥ α £ β §
α + β = π/2
sin α =
n sin β = n sin (π/2 − α) = n cos α α = arctg(n) ≈ 0.93 ≈ 53 o

§
θB
sin θh
= η = 1.28
sin θB
sin θh = 1/n §
tgθB = 1/n
θh
§
n
n
2 −1/2
n = (η − 1) ≈ 1.25
§
η =
sin θh
=
sin θB
sin θh
tgθB √
1+tg2θB =
1/n
1/n
√ 1+(1/n) 2
= 1 + (1/n) 2 ,
¥
h
α0
§
§ α0
α1
α1 → α0
h
k
x0
l0
l1(α1)
α0 α1
x1(α1)
β(α1)
β(α0)
α
β(α)
α1
x0 = htg[β(α0)], l0 = k(α1)tg(α0),
x1(α1) = htg[β(α1)], l1(α1) = k(α1)tg(α1),
x1(α1) − x0 = l1(α1) − l0.

k
k(α1) = h tg[β(α1)] − tg[β(α0)]
tg(α1) − tg(α0)
k = lim [k(α1)] = lim h
α1→α0
α1→α0
tg[β(α1)] − tg[β(α0)]
tg(α1) − tg(α0)
¥
α1
→ α0
′ (tg ◦ β)
= h
tg ′
=
(α0) = h (cos −2 ◦β) · cos 2 ·β ′ (α0) = h cos2 (α0)
cos 2 [β(α0)] β′ (α0)
β β(α) = arcsin[sin(α)/n]
′ β (α0) cos2 [β(α0)]
β ′ =
arcsin ◦ sin
′
=
n
1
1 − sin
n
cos 2 ◦β = 1 − sin 2 ◦β = 1 − sin 2 ◦ arcsin ◦ sin
n
·
2
1/2 1
· cos =
n
= 1 −
cos
n2 2
− sin
,
1/2
2 sin
n
= n2 − sin 2
n2 .
k
k = h
n 2 cos 3 (α0)
[n2 − sin 2 .
(α0)] 3/2
¨¤
§
¥
n
R λ
R
r
d(r)

¥
r
n
λn = λ/n §
∆φ(r) = 2d(r) 2π
λn
− π = 4πd(r)n
λ
£
d(r) ¥
2π ∆φ(r) = 2πm
d(r)
d(r) =
(2m + 1)λ
.
4n
− π,
√
d(r) = R − R2 − r2 R − R2 − r2 (2m + 1)λ
= .
4n
rm =
2m + 1
λR −
2n
2m + 1
4n
π
2 λ2
2m + 1
≈
2n λR,
R ≫ λ
¥¥
¨¤
§
λ
¥
P
¥
R2
UFresnel(L) = C eikL
L
S
R1
P
r2 ik
U(r)e 2L d 2 r,
k = 2π/λ C = k/(2πi) L S U(r)
r IFresnel(L) = |UFresnel(L)| 2

e
UFresnel(L) = CU0
ikL
L
¥
U0
L
e
= CU0
ikL π
L 2
S
e iπ x2 +y 2
e
λL dxdy = CU0
ikL
λL r2
eiπ λL
2πi
= CU0λ
e
2
ikL e iπ R2 2 +R2 1
2λL sin
IFresnel(L) = |UFresnel(L)| 2 =
R2
L
=
R1
CU0eikLλ 4i
2 R2 − R2 1
2λL π
.
CU0λ
2
π
2
R2
r2 iπ
e λL rdrdφ =
0 R1
e iπ R2 2
λL − e iπ R2 1
λL
2 sin 2
2 R2 − R2 1
2λL π
.
§
¥
e |e| = 1
ikL
CU0e
U(e) =
L
e
S
−ikr d 2 r =: C0
S
e −ikr d 2 r
¥¥
k = ke
x
k
θ
φ
z
£
z ¥ k
x z
z θ
x := kr sin θ
U(θ) = C0
U(θ) =
C0
k 2 sin 2 θ
r
R
2π R
e
0 0
−ikr sin θ cos φ rdrdφ.
2π kR sin θ
e
0 0
−ix cos φ xdxdφ.
y
=

¥
1
2π
2π
e
0
−ix cos φ dφ = J0(x).
U(θ) = C02π
k 2 sin 2 θ
z
I(θ) = (|C0|2πR 2 ) 2
0
kR sin θ
0
tJ0(t)dt = zJ1(z).
J0(x)xdx.
U(θ) = C02πR 2 J1(kR sin θ)
.
kR sin θ
J1(kR sin θ)
kR sin θ
2 =: I0
J1(kR sin θ)
kR sin θ
θ = 0 J1(x) ≈ x
¥
Imax = I(0) = (|C0|πR 2 ) 2 = I0
4 .
§
d
d
y
x
2
.
2
x ≪ 1
§
e x y
p q
U(p, q) = C0
a
2
− a
2
a
2a
b −d+ 2
dx
−d− b
dye
2
−ik(px+qy) a
+ dx
−a
b
d+ b
2
d− b
2
b
dye −ik(px+qy) .

U(p, q) = − C0
k2
b
b e ikq
e 2 −ikq
− e 2 ikqd
pq
a
a ikp
e 2 −ikp
− e 2 −ikqd
+ e e ikpa − e −ikpa .
e ax dx = e ax /a
2i sin(x)
e ix − e −ix =
U(p, q) = 4C0
k2 b
sin(kq
pq 2 )
e ikqd sin(kp a
2 ) + e−ikqd
sin(kpa) .
e ix + e −ix = 2 cos(x)
I(e) = I(p, q) = U ∗ (p, q)U(p, q) =
4|C0|
k2 2 sin
pq
2 (kq b
2 )
sin 2 (kp a
¨¤
¡
§
2 ) + sin2 (kpa) + 2 sin(kp a
2
) sin(kpa) cos(2kqd) .
££
¥
T
y ′
n ′ θ ′
= T
y
.
nθ
n n ′
n f = − ′
c
a b
c d
§§
T
θ
¥¥¤¢£¦¥§©¤§¨
y
n n ′
n
θ
¦¤¤¨
d
n
y ′
θ ′
¢¤£¦¥¨§©§¥¨¤
y
n
θ
y ′

d n ′
nθ = nθ y
′ d
y = y + d · tgθ ≈ y + nθ
n ·
y ′
nθ ′
= T
y
=
nθ
1 d
n
0 1
y
.
nθ
θ
α
T =
1 − d
n .
0 1
¦
α
y = y ′
α
n n ′
′ y
y = y α ≈ − R
sin(θ − α)
sin(θ ′ − α)
≈ θ − α
θ ′ − α
n ′ θ ′ = nθ +
T =
θ ′
R
n′
= ,
n
n − n′
R y,
1 0
1
.
n−n ′
R
′ n n
T =
1 0
1
− n−n′
R
¥
det T = 1
lim (T(R)) =
R→∞
1 0
,
0 1
¦

¥ y ′ = y
θ
α
θ ′
y = y ′
′ θ − α = α − θ α ≈ − y
R
R
α
nθ ′ = − 2n
y − nθ,
R
T =
1
−
0
2n
R −1
.
§§ R
¥
T =
1
2n
R
0
−1
.
§
T
1
R1 R2
d
n 1
T = TR2TdTR1 ,

TR1 =
1 0
1
, Td =
1 d
n
0 1
,
1−n
R1
TR2 =
1 0
1
.
¥
¥
T =
1−n
R2
1 + 1−n d
d
n R1
n
1−n
R1R2n (nR1 + nR2 + (1 − n)d) 1 + 1−n d
n R2
f = − 1
T21
=
R1R2n
(n − 1)(nR1 + nR2 + (1 − n)d)
§
R
d = R
T1 =
d
1 d
,
0 1
.
d
££
1
d
d = R
T2 =
T3 =
T4 =
1
2
R
0
−1
.
1 −d
0 1
.
1
−
0
2
R −1
.
T
T = T4T3T2T1 =
−1
0
0
−1
.
2 ¥
T