Chapter 1: Mechanics
3
Newtons 3rd law is given by: Faction = Freaction . For the power P holds: P = W = F v . For the total energy W , the kinetic energy T and the potential energy U holds: W = T + U ; T = U with T = 1 mv 2 . 2 The kick S is given by: S
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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
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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
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
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 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 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 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
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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 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
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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:
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Physics Formulary by ir. J.C.A. Wevers
Keplers orbital equations In a force eld F = kr 2 , the orbits are conic sections with the origin of the force in one of the foci (Keplers 1st law). The equation of the orbit is: r() = with 1 + cos( 0 ) , or: x2 +
Chapter 1: Mechanics
5
1.4.2 Tensor notation
Transformation of the Newtonian equations of motion to x = x (x) gives: dx x dx = ; dt x dt The chain rule gives: d2 x d dx d = = 2 dt dt dt dt so: x dx dt x = dx d x d2 x + dt2 x dt dt x x
2 x dx x dx d x =
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Physics Formulary by ir. J.C.A. Wevers
1.6 Dynamics of rigid bodies
1.6.1 Moment of Inertia
The angular momentum in a moving coordinate system is given by: L = I + Ln where I is the moment of inertia with respect to a central axis, which is given by: I=
Chapter 1: Mechanics
7
the equations of Lagrange can be derived: L d L = dt qi qi When there are additional conditions applying to the variational problem J (u) = 0 of the type K (u) =constant, the new problem becomes: J (u) K (u) = 0.
1.7.2 Hamilton mech
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Physics Formulary by ir. J.C.A. Wevers
If the equation of continuity, t +
( v ) = 0 holds, this can be written as: cfw_ , H + =0 t
For an arbitrary quantity A holds: dA A = cfw_A, H + dt t Liouvilles theorem can than be written as: d = 0 ; or: dt pdq
Chapter 2
Electricity & Magnetism
2.1 The Maxwell equations
The classical electromagnetic eld can be described by the Maxwell equations. Those can be written both as differential and integral equations: (D n )d2 A = Qfree,included (B n )d2 A = 0 E ds = d
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Physics Formulary by ir. J.C.A. Wevers
Here, the freedom remains to apply a gauge transformation. The elds can be derived from the potentials as follows: A E= V , B = A t Further holds the relation: c 2 B = v E .
2.3 Gauge transformations
The potential
Chapter 2: Electricity & Magnetism
11
2.5.2 Electromagnetic waves in matter
The wave equations in matter, with c mat = ()1/2 the lightspeed in matter, are:
2
2 t2 t
E =0,
2
2 t2 t
B=0
give, after substitution of monochromatic plane waves: E = E exp(i(
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
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 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
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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
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
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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 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
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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
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
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