formulaire Optique

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Page 1: formulaire Optique

Produits scalaire et vectoriel

~a ·~b = ‖~a‖ × ‖~b‖ × cos“

~a,~b”

‖~a ∧~b‖ = ‖~a‖ × ‖~b‖ ײ

˛

˛sin“

~a,~b”˛

˛

˛

~a ·“

~b ∧ ~c”

= ~b ·(~c ∧ ~a) = ~c ·“

~a ∧~b”

= det“

~a,~b,~c”

= ±vol“

~a,~b,~c”

~a ∧ (~b ∧ ~c) = ~b × (~a · ~c) − ~c × (~a ·~b)

Systemes de coordonnees orthogonaux

y

z

x

b

b

b

z

y

x

~ey

~ez

~ex

~eθ

~er

r

ρ

r sin θ ~eϕ

~ez

~eρ

~eϕ

ϕ

θ

M

m

O

~ex

~ey

ϕ

ϕ

~eρ

~eϕ

~ez

~eρ

θ

θ

~er

~eθ

ρ = Om > 0 0 6 ϕ < 2π

r = OM > 0 0 6 θ 6 π

~eρ = ~ex cos ϕ + ~ey sin ϕ, ~eϕ = −~ex sin ϕ + ~ey cos ϕ~er = ~ez cos θ + ~eρ sin θ, ~eθ = −~ez sin θ + ~eρ cos θx = ρ cos ϕ, y = ρ sin ϕ, z = r cos θ, ρ = r sin θ

d~r = dx~ex + dy~ey + dz~ez ; dτ = dx × dy × dzd~r = dρ~eρ + ρdϕ~eϕ + dz~ez ; dτ = ρdρ × dϕ × dzd~r = dr~er + rdθ~eθ + r sin θdϕ~eϕ ; dτ = r2 dr × sin θ dθ × dϕ

Operateurs differentiels

dF =−−→grad F · d~r ;

−−→grad F = ~∇F =

∂F

∂x~ex +

∂F

∂y~ey +

∂F

∂z~ez

I

S

~V · d~S =

Z

V

div ~V dτ (Ostrogradski ; S est fermee et delimite

le volume interieur V) ; div ~V = ~∇ · ~V =∂Vx

∂x+

∂Vy

∂y+

∂Vz

∂zI

Γ

~V · d~r =

Z

Σ

−→rot ~V d~S (Stokes ; Γ est fermee et constitue le bord

oriente de Σ) ;−→rot ~V = ~∇∧ ~V =

»

∂Vz

∂y− ∂Vy

∂z

~ex + . . .

∆F = div−−→grad F ; ∆F = ∇2F = ∂2F

∂x2 + ∂2F∂y2 + ∂2F

∂z2

−→rot−→rot ~V =−−→grad div ~V − ∆~V ; ∆~V = ∇2~V = ∆Vx~ex + . . .

d~V =“

d~r · −−→grad”

~V ;“

~a · −−→grad”

~V =“

~a · ~∇”

~V = ax∂~V∂x

+ . . .

Coordonnees cylindro-polaires

−−→grad F =

∂F

∂ρ~eρ +

1

ρ

∂F

∂ϕ~eϕ +

∂F

∂z~ez

div ~V =1

ρ

∂ρ(ρVρ) +

∂Vϕ

∂ϕ

ff

+∂Vz

∂z

−→rot ~V =

1

ρ

∂Vz

∂ϕ− ∂Vϕ

∂z

ff

~eρ +

∂Vρ

∂z− ∂Vz

∂ρ

ff

~eϕ + . . .

. . . +1

ρ

∂ρ(ρVϕ) − ∂Vρ

∂ϕ

ff

~ez

∆F =1

ρ

∂ρ

ρ∂F

∂ρ

«

+1

ρ2

∂2F

∂ϕ2+

∂2F

∂z2

∆F (ρ) = 0 ⇒ F (ρ) = A ln ρ ⇒ ~V =−−→grad F = A~eρ/ρ

Coordonnees spheriques

−−→grad F =

∂F

∂r~er +

1

r

∂F

∂θ~eθ +

1

r sin θ

∂F

∂ϕ~eϕ

div ~V =1

r2

∂r

`

r2Vr

´

+1

r sin θ

∂θ(sin θVθ) +

∂Vϕ

∂ϕ

ff

−→rot ~V =1

r sin θ

∂θ(sin θVϕ) − ∂Vθ

∂ϕ

ff

~er + . . .

. . . +1

r

1

sin θ

∂Vr

∂ϕ− ∂

∂r(rVϕ)

ff

~eθ +1

r

∂r(rVθ) − ∂Vr

∂θ

ff

~eϕ

∆F =1

r2

∂r

r2 ∂F

∂r

«

+1

r2 sin θ

∂θ

sin θ∂F

∂θ

«

+1

r2 sin2 θ

∂2F

∂ϕ2

∆F (r) = 0 ⇒ F (r) = −A/r ⇒ ~V =−−→grad F = A~er/r2

Proprietes generales

−−→grad

“−−→Cte · ~r

=−−→Cte ;

−→rot“−−→Cte ∧ ~r

= 2 ×−−→Cte ; div~r = 3

−→rot“−−→grad F

= 0 ; −→rot ~y = 0 ⇒ ∃x /~y =−−→grad x

div“−→rot ~V

= 0 ; div ~y = 0 ⇒ ∃~x /~y =−→rot ~x

−−→grad (F G) = F

−−→grad G + G

−−→grad F

div“

F ~V”

= F div ~V + ~V · −−→grad F

div“

~U ∧ ~V”

= ~V · −→rot ~U − ~U · −→rot ~V

−→rot“

F ~V”

= F−→rot ~V +

−−→grad F ∧ ~V

−−→grad

~U · ~V”

= ~U∧−→rot ~V +~V ∧−→rot ~U+“

~V · −−→grad”

~U+“

~U · −−→grad”

~V

−→rot“

~U ∧ ~V”

=“

div ~V”

~U −“

div ~U”

~V −“

~U · −−→grad”

~V + . . .

. . . +“

~V · −−→grad”

~U

Theoremes integraux

Γ est fermee et constitue le bord oriente de Σ.

Stokes :

I

Γ

~V · d~r =

Z

Σ

−→rot ~V d~S

Kelvin :

I

Γ

F d~r =

Z

Σ

d~S ∧ −−→grad F

S est fermee et delimite le volume interieur V.

Ostrogradski :

I

S

~V · d~S =

Z

V

div ~V dτ

Gradient :

I

S

F d~S =

Z

V

−−→grad Fdτ

Primitives usuelles

Fonction Primitive

(x − a)n, n6=−1 1

n + 1(x − a)n+1

1

x − aln |x − a|

exp(ax)1

aexp(ax)

ln x x ln x − xcos x sin xsin x − cos xtan x − ln | cos x|

1

tan xln | sin x|

1/ cos2 x tanx

1/ sin2 x − 1

tanx

1/ cos x ln˛

˛

˛tan“x

2+

π

4

”˛

˛

˛

1/ sin x ln˛

˛

˛tan

x

2

˛

˛

˛

ch x sh xsh x ch x1/ ch2 x thx

1/ sh2 x − 1

thx1/ chx 2 arctan (exp(x))

1/ sh x ln˛

˛

˛th

x

2

˛

˛

˛

1

a2 + x2

1

aarctan

x

a1

a2 − x2

1

2aln

˛

˛

˛

˛

a + x

a − x

˛

˛

˛

˛

=1

aargth

x

a

1√a2 + x2

ln

x

a+

r

x2

a2+ 1

!

= argshx

a1√

a2 − x2arcsin

x

a`

1 ± x2´−3/2 x√

1 ± x2

Page 2: formulaire Optique

Fonctions de Bessel

Equation de Bessel : x2y′′ + xy′ + (x2 − ν2)y = 0

Solution generale : y(x) = αJν(x) + βYν(x)

Jν(x) ∼x→0xν

2νν!; Yν(x) ∼x→0 −2ν(ν − 1)!

πxν

Jν+1(x) =2ν

xJν(x) − Jν−1(x), Yν+1(x) =

xYν(x) − Yν−1(x)

dJν

dx=

Jν+1(x) − Jν−1(x)

2,

dYν

dx=

Yν+1(x) − Yν−1(x)

2

Jν(x) =1

π

Z π

0

cos (νθ − x sin θ) dθ

sin (x sin θ) = 2∞X

n=1

J2n−1(x) sin ([2n − 1]θ)

cos (x sin θ) = J0(x) + 2

∞X

n=1

J2n(x) cos (2nθ)

xb b b b b b b b

b

0

1

Jν(x

)

ν = 0ν = 1

ν = 2

x

b

b b

1,2

2

0

1

4J2 1(π

x)

(πx)2

Moments d’inertie de solides pleins

J=

MR

2

2

J=

M“

R2 4

+h2

12

hR

b

R

J = 25MR2

a

b

c

J = M12

`

b2 + c2´

Spectre electromagnetique

400

nm

750

nm

UV

10

nm

X

1pm

γ IR

1m

m

µO

10

cm

radio

400

nm

750

nm

Fonctions de l’Optique

sinc(x) =sin x

x,

Z ∞

−∞

sinc(x) dx =

Z ∞

−∞

sinc2(x) dx = π

x

π

b

b

b b b0

1

sincx

sinc2

x

4 × 10−2

RN (x) =

˛

˛

˛

˛

˛

N−1X

k=0

exp (ikx)

˛

˛

˛

˛

˛

2

=sin2 Nx/2

sin2 x/2

x

πb b b

b

0

1

RN

(x)/

N2

N=

6N

=10

x = π

x = 2πN

˛

˛

˛

˛

˛

∞X

k=0

ρk exp (ikx)

˛

˛

˛

˛

˛

2

=1

(1 − ρ)2Fm(x) si ρ < 1

Fm(x) =1

1 + m sin2(x/2)avec m =

(1 − ρ)2

x

πb b b

b

0

1

F m(x

)

m=

10

m=

100

Classification periodique des elements

metaux

non-metaux

semi-conducteurs

lanthanides

actinides

H He

Li Be B C N O F Ne

Na Mg Al Si P S Cl Ar

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

Cs Ba * Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

Fr Ra + Lr Rf Ha Sg Ns Hs Mt

* La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb

+ Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No

Rayonnement thermique

du

dν=

8πhν3

c3

1

exp“

hνkBT

− 1;

dν=

c

4

du

Z ∞

0

x3dx

exp(x) − 1=

π4

15, σ =

2π5k4B

15c2h3

λmaxT = CW = 0, 201hc

kB= 2, 90 × 10−3 m · K

0, 98σT 4

dλ=

2πhc2

λ5

1

exp“

hcλkBT

− 1

Constantes fondamentales

c = 3, 00 × 108 m · s−1 me = 9, 11 × 10−31 kge = 1, 60 × 10−19 C mp ≃ mn ≃ 1, 67 × 10−27 kgǫ0 = 8, 85×10−12 F·m−1 µ0 = 4 × π × 10−7 H · m−1

F = 96 500 C · mol−1 NA = 6, 02 × 1023 mol−1

R = 8, 31 J · K−1 · mol−1 G = 6, 67 × 10−11 m3 · kg−1 ·h = 6, 63 × 10−34 J · s σ = 5, 67 × 10−8 W · m−2 · KkB = 1, 38×10−23 J·K−1 TT = 273, 16 K

Donnees astronomiques

M⊙ = 1, 99 × 1030 kg R⊙ = 6, 96 × 108 m1UA = 1, 50 × 1011 m 1AL = 9, 46 × 1015 m1pc = 3, 09 × 1016 m 1 j (solaire) = 86 400 s1 an = 365, 25 j (solaire) 1 j (sideral) = 86 164 s

Terre Lune

M = 5, 98 × 1024 kg M = 7, 35 × 1022 kgR = 6, 38 × 106 m R = 1, 74 × 106 md⊙ = 1UA dTerre = 3, 84 × 108 me = 0, 017 e = 0, 055T = 1an T = 27, 3 j (solaire)

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