Formulário
Cálculo e dados numéricos
kB = 1, 38 × 10−23 J/K ; h = 6, 63 × 10−34 Js ; R = 8, 31J/K · mol ;
NA = 6, 02 × 1023 mol−1 ; c = 3, 0 × 108 m/s ; 1eV = 1, 6 × 10−19 J ;
ln n! = n ln n − n + O(ln n) ; Γ(n) = (n − 1)! ; Γ(n) = (n − 1)Γ(n − 1) ; Γ(1/2) =
R
R +∞
2 dx = π 1/2 ; +∞ xn exp −ax2 dx = 1 a−(n+1)/2 Γ (n+1)
exp
−ax
a
2
2
0
−∞
R
R
x2 + . . .
udv = uv − vdu ; (1 + x)n = 1 + nx + n(n−1)
2
ex −e−x
2
; cosh(x) =
d
dx (senh(x)) = cosh(x) ;
2
d
dx (tgh(x)) = (sech(x)) ;
senh(x) = x +
π
Vesf era = 43 πr3
Aesf era = 4πr2
senh(x) =
√
x3
3!
ex +e−x
2
; tgh(x) =
senh(x)
cosh(x)
d
dx (cosh(x)) = senh(x);
2
d
dx (cotgh(x)) = −(cossech(x))
+ . . . ; cosh(x) = 1 +
x2
2!
+ . . . ; tgh(x) = x −
x3
3
+ . . . ; cotgh(x) =
1
x
+
x
3
+ ...
Mecânica, Eletromagnetismo etc
P~
F = m~a
F = −kx
Ec = 12 Iω 2
W =
~ · d~l
F
U = 21 kx2
R
I = r2 dm
∆U = −Wcons
~ =
F
R
1 qQ
4πǫ0 r2 r̂
~ =0
∇×E
~ M = q~v × B
~
F
R
~ = µ0 I d~l′ ×2R̂
B
4π
R
P
ω=
E = Ec + U
q
W = ∆Ec
k
m
T = 2π
E = Ec + U
I
κ
I = ICM + mr2
τ = Iα
2
Fc = mvr
H
~a
~ · d~
ΦE = E
~ =F
~ /q
E
R
~ · d~l
∆V = − E
∆U = −Wcons
q
~ = −∇V
E
Ep,grav = mgh
~ =
∇·E
V =
ρ
ǫ0
1 q
4πǫ0 r
~ M = Id~l × B
~
dF
I = ∆Q
∆t
H
R
~ · d~l = µ0 IEN V
B
IEN V = ~J · d~a
ρ=
ΦE =
m
V
Qint
ǫ0
∆U = Q∆V
~ = µ0 ~J
∇×B
~ =0
∇·B
Fı́sica Estatı́stica e Termodinâmica
S = kB ln Ω
ln Ω
N
s = limN →∞
∆U = Q − W
∂u
cv = ∂T
=T
v
dU = T dS − P dV
dS = dQ
T
∂s
∂u
∂s
cP = ∂T
= T ∂T
∂T v
P
P
Tc = TK − 273, 15
R = NA k B
H = U + PV
∂v
β = v1 ∂T
P
Q = C∆T
κ = − v1
∂v
∂P T
Q = mc∆T
P vM = RT (vM volume molar), P V = N kB T ou P v = kB T
P −βE
∂ ln ZN
−βEj
j
; β = k 1T
Pj = e
/Z ; Z = j e
;U =−
∂β
B
F = −kB T ln Z ; F = U − T S ; G = U − T S + P V ; U = T S − P V + µN
dF = −SdT − P dV + µdN ; df = −sdT − P dv (N fixo)
∂m
∂H T
dF = −SdT − M dH + µdN ; df = −sdT − mdH (N fixo) ; χ =
ZN = (Z1 )N
ZN = Z1N /N !
H = H0 + φpj 2 → hφpj 2 i = 12 kB T
∂H
hxi ∂x
i = δij kB T
j
g = u − T s + P v ; dg = −sdT + vdP ; dG = −SdT + V dP + µdN ; dµ = −sdT + vdP
dP
l
dT = T ∆v
∂V
∂S
=
−
∂P T
∂T P
Z=
ǫ~k =
P
;
∂S
∂V T
m exp [−β(Em
h̄2 k2
2m
, ~k =
=
∂P
∂T V
− µNm )]
2πn1 2πn2 2πn3
L , L , L
;
∂T
∂V S
=−
∂P
∂S V
P V = kB T ln Z
;
∂T
∂P S
Z=
∂V
∂S P
N
N x ZN
=
P
x = eβµ
3/2 P
R
D(ǫ) = Cǫ1/2 , C = 2π 2m
, ~k → V 3 d3 k em três dimensões
h2
(2π)
P
ln Z = ± j ln 1 ± eβ(µ−ǫ) , + para férmions, − para bósons
Z
hnj i = − β1 ∂ln
∂ǫ
j
1
eβ(ǫ−µ) ±1
, + para férmions, − para bósons
i
h
R ∞ xn−1 dx
ξn
1
1
π2 1
Fn (ξ) = Γ(n)
+
o
,ξ ≫ 1
1
+
n(n
−
1)
=
6 ξ2
0 ex−ξ +1
Γ(n+1)
ξ4
f (ǫ) =
gn (1) =
R ∞ xn−1 dx
1
ex −1
Γ(n) 0
ζ(3/2) = 2.612 , ζ(2) =
= ζ(n) (zeta de Riemann) ;
π2
6
, ζ(5/2) = 1.341 , ζ(3) = 1.202 , ζ(7/2) = 1.127 , ζ(4) =
π4
90