Chapter 9: Transport phenomena
43
and the denitions v x = / y , vy = / x holds: AB = (B ) (A). In general holds: 2 2 + = z 2 x y2 In polar coordinates holds: 1 1 = , v = = r r r r Q For source ows with power Q in (x, y ) = (0, 0) holds: = ln(r) so that vr
Chapter 9: Transport phenomena
41
From this one can derive the Navier-Stokes equations for an incompressible, viscous and heat-conducting medium: divv v + (v t C T + C (v t )v )T = = = 0 g gradp +
2 2
v
T + 2 D : D
with C the thermal heat capacity. The f
40
Physics Formulary by ir. J.C.A. Wevers
When the ow velocity is v at position r holds on position r + dr : v (dr ) = v (r )
translation
T can be split in a part pI representing the normal tensions and a part T representing the shear stresses: T = T + pI
Chapter 9
Transport phenomena
9.1 Mathematical introduction
An important relation is: if X is a quantity of a volume element which travels from position r to r + dr in a time dt, the total differential dX is then given by: dX = X X X dX X X X X X dx + dy
38
Physics Formulary by ir. J.C.A. Wevers
8.12 Statistical basis for thermodynamics
The number of possibilities P to distribute N particles on n possible energy levels, each with a g -fold degeneracy is called the thermodynamic probability and is given by
Chapter 8: Thermodynamics
37
8.9 Thermodynamic potential
When the number of particles within a system changes this number becomes a third quantity of state. Because addition of matter usually takes place at constant p and T , G is the relevant quantity. I
36
Physics Formulary by ir. J.C.A. Wevers
3. Isothermic compression at T 2 , removing Q 2 from the system. 4. Adiabatic compression to T 1 . The efciency for Carnots process is: =1 T2 |Q2 | =1 := C |Q1 | T1
The Carnot efciency C is the maximal efciency at
Chapter 8: Thermodynamics
35
From this one can derive Maxwells relations: T V = p S ,
V
S
T p
=
S
V S
,
p
p T
=
V
S V
,
T
V T
p
=
S p
T
From the total differential and the denitions of C V and Cp it can be derived that: T dS = CV dT + T For an ideal gas a
34
Physics Formulary by ir. J.C.A. Wevers
For an ideal gas holds: C mp CmV = R. Further, if the temperature is high enough to thermalize all internal rotational and vibrational degrees of freedom, holds: C V = 1 sR. Hence Cp = 1 (s + 2)R. For their ratio
Chapter 8
Thermodynamics
8.1 Mathematical introduction
If there exists a relation f (x, y, z ) = 0 between 3 variables, one can write: x = x(y, z ), y = y (x, z ) and z = z (x, y ). The total differential dz of z is than given by: dz = z x dx +
y
z y
dy
x
32
Physics Formulary by ir. J.C.A. Wevers
7.5 Collisions between molecules
The collision probability of a particle in a gas that is translated over a distance dx is given by n dx, where is v1 2 2 with u = v1 + v2 the relative velocity between the cross se
Chapter 7: Statistical physics
31
7.3 Pressure on a wall
The number of molecules that collides with a wall with surface A within a time is given by: dN=
3 2
nAv cos()P (v, , )dvdd
0
0
0 1 4n
From this follows for the particle ux on the wall: = d3 p =
v .
Chapter 7
Statistical physics
7.1 Degrees of freedom
A molecule consisting of n atoms has s = 3n degrees of freedom. There are 3 translational degrees of freedom, a linear molecule has s = 3n 5 vibrational degrees of freedom and a non-linear molecule s =
Chapter 6: Optics
29
in the plane through the transmission direction and the optical axis. Dichroism is caused by a different absorption of the ordinary and extraordinary wave in some materials. Double images occur when the incident ray makes an angle wit
28
Physics Formulary by ir. J.C.A. Wevers
The dispersion of a prism is dened by: d d dn = d dn d where the rst factor depends on the shape and the second on the composition of the prism. For the rst factor follows: 2 sin( 1 ) d 2 = dn cos( 1 (min + ) 2 D=
Chapter 6: Optics
27
with I = |S | it follows: R + T = 1. A special case is r = 0. This happens if the angle between the reected and transmitted rays is 90 . From Snells law it then follows: tan( i ) = n. This angle is called Brewsters angle. The situatio
26
Physics Formulary by ir. J.C.A. Wevers
6.3 Matrix methods
A light ray can be described by a vector (n, y ) with the angle with the optical axis and y the distance to the optical axis. The change of a light ray interacting with an optical system can be
Chapter 6: Optics
25
D := 1/f is called the dioptric power of a lens. For a lens with thickness d and diameter D holds to a good approximation: 1/f = 8(n 1)d/D 2 . For two lenses placed on a line with distance d holds: 1 1 1 d = + f f1 f2 f1 f2 In these e
Chapter 6
Optics
6.1 The bending of light
For the refraction at a surface holds: n i sin(i ) = nt sin(t ) where n is the refractive index of the material. Snells law is: n2 1 v1 = = n1 2 v2 If n 1, the change in phase of the light is = 0, if n > 1 holds:
Chapter 5: Waves
23
3. Ez and Bz are zero everywhere: the Transversal electromagnetic mode (TEM). Than holds: k = and vf = vg , just as if here were no waveguide. Further k I , so there exists no cut-off R frequency. In a rectangular, 3 dimensional resona
22
Physics Formulary by ir. J.C.A. Wevers
5.4 Green functions for the initial-value problem
This method is preferable if the solutions deviate much from the stationary solutions, like point-like excitations. Starting with the wave equation in one dimensio
Chapter 5: Waves
21
The equation for a harmonic traveling plane wave is: u(x, t) = u cos( k x t + )
If waves reect at the end of a spring this will result in a change in phase. A xed end gives a phase change of /2 to the reected wave, with boundary condi
Chapter 5
Waves
5.1 The wave equation
The general form of the wave equation is: 2u = 0, or:
2
u
1 2u 2u 2u 2u 1 2u = + 2 + 2 2 2 =0 2 t2 2 v x y z v t
where u is the disturbance and v the propagation velocity. In general holds: v = f . By denition holds:
Chapter 4: Oscillations
19
1. Series connection: V = IZ , Ztot =
i
Zi , Ltot =
i
Li ,
1 = Ctot
i
1 Z0 , Z = R(1 + iQ ) , Q= Ci R
2. parallel connection: V = IZ , 1 = Ztot Here, Z0 = 1 1 , = Zi Ltot 1 , Ctot = Li Ci , Q =
i
i
i
R R , Z= Z0 1 + iQ
L 1 . and
Chapter 4
Oscillations
4.1 Harmonic oscillations
The general form of a harmonic oscillation is: (t) = ei(t) cos( t ), where is the amplitude. A superposition of several harmonic oscillations with the same frequency results in another harmonic oscillation
Chapter 3: Relativity
17
3.2.4 The trajectory of a photon
For the trajectory of a photon (and for each particle with zero restmass) holds ds 2 = 0. Substituting the external Schwarzschild metric results in the following orbital equation: du d d2 u + u 3mu
16
Physics Formulary by ir. J.C.A. Wevers
r > 2 m:
u v u v
= =
r r 1 exp cosh 2m 4m r r 1 exp sinh 2m 4m 1 1 r r exp sinh 2m 4m r r exp cosh 2m 4m
t 4m t 4m t 4m t 4m
r < 2 m:
= =
r = 2m: here, the Kruskal coordinates are singular, which is necessary
14
Physics Formulary by ir. J.C.A. Wevers
W 2 = m2 c4 + p2 c2 . p = mr v = m0 v = W v/c2 , and pc = W where = v/c. The force is dened by 0 F = dp/dt. 4-vectors have the property that their modulus is independent of the observer: their components can chang
Chapter 3
Relativity
3.1 Special relativity
3.1.1 The Lorentz transformation
The Lorentz transformation (x , t ) = (x (x, t), t (x, t) leaves the wave equation invariant if c is invariant: 2 2 1 2 2 2 2 1 2 2 + 2+ 2 2 2= + + 2 2 x2 y z c t x 2 y 2 z 2 c t
12
Physics Formulary by ir. J.C.A. Wevers
d . If the current If the ux enclosed by a conductor changes this results in an induced voltage V ind = N dt owing through a conductor changes, this results in a self-inductance which opposes the original change: