# Formelsammlung

Werbung
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ππ
ππ
πΌπΌ
Formulas in physics
ππππ
ππππ
ππππ
πΌπΌ =
ππππ
Oscillations and waves ………………………………………….4
Geometrical optics ………………………………………………..5
Thermodynamics …………………………………………………..7
Atomphysics ………………………………………………………….9
Constants ……………………………………………………………..10
π₯π₯π₯π₯
π₯π₯π₯π₯
π₯π₯ = π£π£ ⋅ π‘π‘
displacement [m]
Mean velocity [m/s]
Movement on a circuit with
constant acceleration
π£π£ 2
ππ
2ππππ 1 2ππ
= =
π£π£
ππ
ππ
Angular acceleration
Centripetal acceleration
Vb: speed on the cicuit
Uniform movement on a
circuit
centripetal acceleration a
Periode T
Frequency f
1
πΈπΈππππππ = πΌπΌππ2
2
Energy of rotation
πΌπΌπ΄π΄ = πΌπΌππ + ππππ 2
Theorem of Steiner
πΏπΏοΏ½β = ππβ &times; ππβ
angular momentum
οΏ½οΏ½β = ππβ &times; πΉπΉβ
ππ
Moment of force
Orbital angular momentum
distance [m]
velocity [m/s]
time [s]
πΏπΏ = πΌπΌ ⋅ ππ
distance x [m]
acceleration a [m/s^2]
ππ =
ππππ
= πΌπΌ ⋅ πΌπΌ
ππππ
angular momentum of a
solid
Equation of circular motion
Force and Movement
Mean acceleration [m/s^2]
πΉπΉβ = ππ ⋅ ππβ
Moment in two and three dimensions
ππβ = π₯π₯πποΏ½οΏ½οΏ½β1 + π¦π¦πποΏ½οΏ½οΏ½οΏ½β
οΏ½οΏ½οΏ½βπ§π§
π¦π¦ + π§π§ππ
1
ππ = ππ0 + ππ ⋅ π‘π‘ + πΌπΌπ‘π‘ 2
2
π£π£ 2
ππ =
ππ
ππ =
Moment with constant acceleration
1
π₯π₯ = π₯π₯0 + π£π£0 ⋅ π‘π‘ + πππ‘π‘ 2
2
π£π£(π‘π‘) = π£π£0 + ππ ⋅ π‘π‘
π£π£ 2 = π£π£02 + 2a(π₯π₯ − π₯π₯0 )
1
π₯π₯ − π₯π₯0 = (π£π£0 + π£π£)π‘π‘
2
π₯π₯π₯π₯
ππΜ =
π₯π₯π₯π₯
Angular speed
ππ =
Movement in one dimension
Angular speed and
tangential speed
ππ =
π£π£ππ = ππ ⋅ ππ
Mechanics
π£π£Μ =
π£π£ = ππ ⋅ ππ
Electronics …………………………………………………………….3
π₯π₯π₯π₯ = π₯π₯2 − π₯π₯1
π π
ππ
ππ =
Mechanis ………………………………………………………………1
angular acceleration
Reibung
vector
πππ π ,πππππ₯π₯ = πππ π  ππ
displacement
π₯π₯ππβ = οΏ½οΏ½οΏ½β
ππ2 − οΏ½οΏ½οΏ½β
ππ1
throw
1
π¦π¦ − π¦π¦0 = (π£π£0 π π π π π π π©π©0 )π‘π‘ − πππ‘π‘ 2
2
ππππ 2
2
π£π£0
Horizontal reach
ππ
ππππ = ππππ ππ
Resistance of fluid
1
πΉπΉ = πΆπΆπ€π€ πππππ£π£ 2
2
Movement on a circle
1
π£π£π‘π‘ = οΏ½
2πΉπΉππ
πΆπΆπ€π€ ππππ
2. axiom of Newton
F [N]
Maximal friction
N: Normal force
Dynamic friction
πΆπΆπ€π€ : constant
ππ: density of the fluide
A: front surface
v: speed
π£π£π‘π‘ Maximal speed
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πΉπΉ = ππ
π£π£ 2
ππ
momentum
Centripetal force
Equation of motion
Mechanical Energy
πΈπΈππ =
1
πππ£π£ 2
2
Kinetic energy
kinetic energy
ππ = πΉπΉβ ⋅ ππβ
Golden rule of mechnanics
W: work
F: force
r: distance
πΈπΈππ = πΈπΈππ + ππ
Change of energy
Ef: final energy
Ei: initial energy
W: work
πΉπΉ = −ππ ⋅ π₯π₯
Hook's law
k: spring constant
ππ = πΉπΉππ ⋅ π₯π₯ = ππ ⋅ ππ ⋅ π₯π₯
Friction work
ππ1,2 = ∫ πΉπΉβ ⋅ οΏ½οΏ½οΏ½οΏ½β
ππππ
Generall math formulation
πΈπΈ =
1 2
πππ₯π₯
2
ππππ
ππππ
ππ
ππΜ =
= πΉπΉ ⋅ π£π£
π‘π‘
ππ =
P: Momentum
M: mass
Total momentum
Spring energy
Instantaneous power [W]
Mean power
Potential energy
π₯π₯π₯π₯ = ππππ(π¦π¦ππ − π¦π¦ππ )πΈπΈππππππ
= ππππβ
1
ππ(π₯π₯) = πππ₯π₯ 2
2
πΉπΉ(π₯π₯) =
−ππππ(π₯π₯)
ππππ
Potential energy
Elastic energy
Force dermined from a
potiental
Collisions
J-Integral
Fm: mean force
J-Integral == change of
momentum
Impulse p or
π½π½ = πΉπΉππ ⋅ π₯π₯π₯π₯
π½π½β = π₯π₯ππβ
ππβ = ππ ⋅ π£π£β
2
πΉπΉ =
πΈπΈππππππ =
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β
πππ‘π‘π‘π‘π‘π‘ = ∑
ππ
οΏ½οΏ½οΏ½βπ€π€
π₯π₯π₯π₯
π₯π₯π₯π₯
ππ2
2 ⋅ ππ
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Elektrotechnics
Sources
ππππ Open circuit voltage
πππΎπΎ terminal voltage
ππππ Internal resistance
Electrical charge
1 β£ππ1 β£β£ππ2 β£
πΉπΉ =
4ππππ0 ππ 2
Electrical field (static)
πΈπΈοΏ½β =
πΈπΈ =
πΈπΈ =
Current
Coulombs law
q charges
r distance between charges
Coulomb-Gesetz
ππ0 = 8.85 ⋅ 10−12 πΆπΆ 2 ⁄ππ ⋅ ππ2
ππππ Power of the consumer
RV: Resistance of the
consumer
Ri: internal resistance of
the source
F Force
πΉπΉβ
ππ0
1 β£ππβ£β£
4ππππ0 ππ 2
E-Field of a point charge
1 ππ
ππ = ππππ
2ππππ0 π§π§ 3
E-Field of a dipole
ππ = ππ β πΌπΌ
Ohms law
U tension [V]
R Ohm’s resistance
I Current
πΉπΉβ = πππΈπΈοΏ½β
ππ = ππ β
πΏπΏ
π΄π΄
Force of a charge q in the EField
Specific resistance ππ
L Length
A Area
1
πππ‘π‘ = οΏ½οΏ½ οΏ½
ππππ
−1
ππ
πππ‘π‘ = οΏ½ ππππ
Parallel resistances
Serial resistances
ππ
Condensor
Capacity
πͺπͺππ = οΏ½ πͺπͺππ
ππ
Ct = οΏ½οΏ½
ππ
1
οΏ½
πΆπΆππ
−1
Parallel condensors
Serial condensors
3
πππΎπΎ = ππππ − πΌπΌ ⋅ ππππ
ππ1 ππ1
=
ππ2 ππ2
ππππ = ππππ ⋅
ππππ2
(ππππ + ππππ )2
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Oscillations
πΎπΎ =
Oscillations without
friction
π₯π₯Μ + ππ02 π₯π₯ = 0
2ππ
ππ =
ππ
ππ =
1
ππ
Harmonic oscillations
πΈπΈ(π‘π‘) = πΈπΈ0 ⋅ ππ −2πΎπΎπΎπΎ
vmax = xm ππ
ππ = οΏ½
ππ
ππ
ππ
ππ = 2πποΏ½
ππ
ππ = 2πποΏ½
πΏπΏ
ππ
ππ = 2πποΏ½
πΏπΏππ =
πΌπΌππππ + ππππ 2
ππππππ
πΌπΌππππ + ππππ 2
ππππ
πππ‘π‘ = οΏ½
Maximal speed
Maximal acceleration
Damping term
M: mass
b: Ns/m: Fr = - b v
Damped frequency
Energy loss
Doppler shift
Periode T
ππ′ = ππ ⋅
Frequence [Hz]
ππππππππ = ππ2 π₯π₯ππ
t
ππ2 = ππ02 − πΎπΎ 2
Equation of oscillations
π₯π₯(π‘π‘) = π₯π₯ππ sin(ππππ − ππ0 )
ππ
2ππ
π£π£
Resonance
=
1 &plusmn; π£π£ ⁄ππ
1 β π£π£ ⁄ππ
f, f': Frequency
v: velocity Sender / recipient
c: velocity of sound
Resonance Equation
π₯π₯Μ + 2πΎπΎπ₯π₯Μ + ππ02 = ππππ cos(ππππ)
Spring oscillations
ππ Angular velocity
Stationary solution
Periode of a spring oscillator
Periode
Mathematical pedulum
Periode
Physical pendulum
Amplitude π΄π΄(ππ) =
Reduced length of pendulum
Lr
Phase
πΏπΏ(ππ)
πΉπΉππ ⁄ππ
οΏ½(ππ02 −ππ2 )2 +ππ 2 ππ2 οΏ½ππ2
= arctan οΏ½
πΌπΌππππ
ππ
ππ
=
Damped oscillations
Equation of damped harmonic oscillations
π₯π₯Μ
+
+
=
Μ + 2 πΎπΎ xΜ + ππ02 x = 0
x(t)
4
ππππ
οΏ½
ππ(ππ02 − ππ 2 )
ππππ = οΏ½ππ02 −
ππ 2
2m2
πππππππππ‘π‘ = √4ππππ
π₯π₯(t) = π΄π΄(ππ)cos(ππππ − πΏπΏ(ππ))
ππ
πΎπΎ =
2ππ
ππ2 = ππ02 − πΎπΎ 2
πΈπΈ(π‘π‘) = πΈπΈ0 ⋅ ππ −2πΎπΎπΎπΎ
Resonance frequency
Critical damping
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Waves
ππ =
2ππ
ππ
Wave number
ππ =
2ππ
ππ
Periode
ππ = 2ππππ
Angular frequency
ππ
ππ
Speed in the rope
π£π£ = ππ ⋅ ππ
π£π£ = οΏ½
ππππ =
1
2
ππππππ2 ⋅ π¦π¦ππ
2
Speed of phase
Power
Stationary waves
ππ =
2ππ
ππ
ππππ = ππ ⋅
ππ
2L
wave number
Eigenfrequenzen
n = 1,2,3, …
5
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[P] = W
Geometrical Optics
Besselsche Methode
Mirrors
f&uuml;r Sammellinse
p: distance of object
i: distance of image
f: focal length
a: Abstand Objekt –
Bild
e: Abstand der beiden
Linsenpositionen
f: Brennweite
Refraction
n =
F&uuml;r Streulinse
Refraction index n
c: speed of light in
vacuum
cm: speed of light in
medium
c
cm
f1: Brennweiter Streulinse
a: Distanz ObjektStreulinse
d: Distanz StreulinseSammellinse
b: Distanz: SammellinseBild
Law of refaction
Sign convention
r &lt; 0 for concave
surface
r &gt; 0 f&uuml;r convex
surface
Dioptry D
Korrektur Kurzsichtig
smax: maximale
Sehdistanz
Lens maker formula
1
= (n-1)οΏ½r 1
1
r2
οΏ½
Abbildungsgleichung
d&uuml;nner Linsen
Intensit&auml;t bei
isotropem
Leuchtk&ouml;rper
[I] = W/m2
ππ ⋅ sin(πΌπΌ) = ππππππππππ
ππ =
−π΅π΅ −ππ
=
πΊπΊ
ππ
1 1 1
+ =
ππ ππ ππ
Korrektur Weitsichtig πΌπΌ(ππ) = ππ
2
ri &lt; 0 for convave
ri &gt; 0 for convex
f
ππ0
ππππ
Linse in Serie
D&uuml;nne Linsen in Serie
ππ1 ππ2 ππ2 − ππ1
+
=
ππ
ππ
ππ
Refraction on a
spherical surface
1
ππππ =
s0: normale
Sehdistanz
smin: minimale
Sehdistanz
4ππππ
smin &gt; s0 f&uuml;r
Weitsichtigkeit
1 1 1
+ =
ππ ππ ππ
ππ = 2 ⋅ ππ
Optische
Instrumente
Konvex f &lt; 0
konkav f &gt; 0
6
ππππ =
ππ0
ππππ
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Lupe
s0 = 25 cm
f Brennweite der Lupe
Beamexpander
D:
Ausgangsdurchmesse
r
d:
Eingangsdurchmesser
f1: Eingangslinse
f2: Ausgangslinse
ππ = 2οΏ½
ππ2
− ππ ⋅ ππ
4
ππ2 − ππ 2
4 ⋅ ππ
1 1
1
= +
ππ1 ππ ππ − 1
1 1
−
ππ2 ππ
ππ =
ππππππππ
Fernrohr
ππ =
ππππππ
m:
Winkelvergr&ouml;sserung
fok: Okularlinse
fobj: Objektivlinse
Mikroskop
s0 = 25 cm
s = Tubusl&auml;nge
fok = Okularbrennweite
fobj = Objektivbrennweite
ππ =
π₯π₯π₯π₯ =
−π΅π΅ −ππ
=
πΊπΊ
ππ
π·π· =
1
1
ππ
=
−1
π π ππππππ
ππππππππππππππ
π₯π₯π₯π₯ &lt; 0
1
1
π₯π₯π₯π₯ = −
π π 0 π π ππππππ
π₯π₯π₯π₯ &gt; 0
ππ = 2οΏ½
ππ =
ππ2
− ππ ⋅ ππ
4
ππ2 − ππ 2
4 ⋅ ππ
7
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Thermodynamics
π₯π₯πΏπΏ = πΌπΌ ⋅ πΏπΏ ⋅ π₯π₯ππ
π₯π₯π΄π΄ = π½π½ ⋅ π΄π΄ ⋅ π₯π₯π₯π₯
π½π½ = 2 ⋅ πΌπΌ
π₯π₯π₯π₯ = πΎπΎ ⋅ ππ ⋅ π₯π₯π₯π₯
πΎπΎ = 3 ⋅ πΌπΌ
[Rs] = J/kg/K
[v] = m3/kg
M: mol volume
v: spezific volume
vm: molar volume
1.Dimensions
2 dimensions
Equation of real gas
3 dimensions
ππ ⋅ ππ = ππ ⋅ πππ π  ⋅ ππ
π£π£ππ = ππ ⋅ π£π£
ππ = πππ π  ⋅ ππ
Q = m C βT
Specific heat
capacity
[C] = J kg/K
NA = 6.022&middot;1023/mol
ππ = ππ ⋅ ππππ ⋅ π₯π₯ππ
Molar heat
[cm] = J/mol/K
[n] mol-1
Work Wx
πΆπΆΜ =
ππ2
∫ππ1
πΆπΆ(ππ)ππππ
ππ2 − ππ1
ππ = πΏπΏ ⋅ ππ
πππππ£π£ = ππ(ππ) ⋅ ππππ
πππππ£π£ = ππ(π£π£)ππππ
ππππππ = −ππ(ππ) ⋅ ππππ
ππππππ = −π£π£(ππ) ⋅ ππππ
Mean value
ππππ
ππππ
Melting enthalpy
[L] = J/kg
ππ ⋅ π£π£ = πππ π  ππ
ππ ⋅ π£π£ππ = ππ ⋅ ππ
V-work Wv
p-work
Heat
[Q] = J
[q] = J/kg
(Spec.) Inner Energy
ππ
[U] = J
⋅ πππ΄π΄ ⋅ πππ΅π΅ ⋅ ππ
[u] = J/kg
2
ππ = πππ΄π΄ ⋅ πππ΅π΅
Perfect gas: f = 3
ππ =
Perfect gas
ππ ⋅ ππ = ππ ⋅ πππ π  ⋅ ππ
Real gas factor Z
πΏπΏπΏπΏ
ππ
πΏπΏππ
ππππ =
ππ
ππππ =
Equation of equilibrium
of the perfect gas.
[V] = m3
[p] = Pa
[m] = kg
R = universelle
R = 8.314 J/mol/K
(spez.) Entropy S, s
[S] = J/K
[s] = J/K/kg
4 Principles of thermodynamics
8
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Four elementary Processes in
closed systems
Closed systems
ππππππ = −ππ(ππ) ⋅ ππππ Volume work
ππππππ = −π£π£(ππ) ⋅ ππππ
Isochor process
1a. closed systems
ππππ = πΏπΏπΏπΏ − πΏπΏππππ
ππππ = πΏπΏπΏπΏ − πΏπΏπ€π€π£π£
V = const
πππππ£π£ = ππ(ππ) ⋅ ππππ
πππ€π€π£π£ = ππ(π£π£)ππππ
Work
Conversion of energy
ππ
= ππππππππππ
ππ
Equation of state
1b. open Systems
heat
ππππ π₯π₯ = πΏπΏπΏπΏ − πΏπΏππππ
ππβ π₯π₯ = πΏπΏπΏπΏ − πΏπΏπ€π€ππ
v-work
isobar process
2. natural processes ππππ = πΏπΏπΏπΏ
ππ
takes the direction to
πΏπΏπΏπΏ
ππππ =
increase the total
ππ
entropy.
ππ12 = π’π’2 − π’π’1
π€π€π£π£ = 0
p = const
π£π£
= ππππππππππ
ππ
Equation of state
heat
ππππ &gt; 0
ππππ &gt; 0
v-work
ππ12 = β2 − β1
π€π€π£π£12 = ππ ⋅ (ππ2 − ππ1 )
isotherm process T = const
lim π₯π₯π₯π₯ = 0
3. It is not possible to ππ→0
lim π₯π₯π₯π₯ = 0
hit the absolut zero
ππ→0
temperature by a
natural process.
Equation of state
heat
v-work
0. Two systems in a
thermodynamical
process have the
same temperature.
ππ
= ππππππππππ
ππ
ππ ⋅ π£π£ = πππππππππ‘π‘
π€π€π£π£12 = ππ1 ⋅ π£π£1 ⋅ ln
ππ12 = ππ ⋅ ππ ⋅ ln
π£π£2
π£π£1
exponent
k = cp/cv = 1.4
Equation of state
ππ ⋅ π£π£ ππ = ππππππππππ
ππ ⋅ π£π£ ππ = πππππππππ‘π‘
ππ ⋅ π£π£ ππ = πππππππππ‘π‘
open systems
9
π£π£2
π£π£1
ππ ⋅ π£π£ ππ = πππππππππ‘π‘
π»π» = ππ + ππ ⋅ ππ
β = π’π’ + ππ ⋅ π£π£
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Enthalpy H, h
[H] = J
[h] = J/kg
Totale Enthalpy H*
[H*] = J
[h*] = J/kg
1
π»π» π₯π₯ = π»π» + ππππ 2
2
1
π»π» π₯π₯ = ππ + ππππ + ππππ 2
2
1
β π₯π₯ = β + ππ 2
2
1
β π₯π₯ = π’π’ + ππ ⋅ π£π£ + ππ 2
2
10
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Quantum
mechanics
Quantisierung der
Energieniveaus
h: Planckkonstante
f: Photonenfrequenz
Eu: oberes
Energieniveau
El: unteres
Energieniveau
Projektion des
Spindipolmoment
Photoelektrischer
Effekt
β ⋅ ππ = πΈπΈπ’π’ − πΈπΈππ
Photonenenergie
f: Frequenz
Wellenl&auml;nge Frequenz
Photonenimpuls
Photonenmasse
Atomarer Drehimpuls
(Bahndrehimpuls)
Klassischer Drehimpuls
Atomarer
Bahndrehimpuls
L Bahndrehimpuls
l Quantenzahl, ππππππ, ππ &lt;
ππ
Photelektrische
Gleichung
πΏπΏ = ππ ⋅ ππ ⋅ π£π£
πΏπΏ = οΏ½ππ(ππ + 1) ⋅ β
K: Kinetische Energie
der
Austrittselektronen
ππππ Austrittsarbeit
V: Stoppotential
n: Hauptquantenzahl
Z-Projektion des
Drehimpulses,
β£πππ§π§ β£ ≤ ππ, πππ§π§ ππππ
Bohrsches
Magnetonπππ΅π΅
Magnetisches
Dipolmoment
Z-Projektion des
magnetischen
Dipolmoments
πΏπΏπ§π§ = ππππ ⋅ β
πππ΅π΅ =
πππ΅π΅ =
οΏ½οΏ½οΏ½οΏ½β
Materiewellen –
Wahrscheinlichkeit
ππ
β
2ππππ
De Broglie
Wellenl&auml;nge
−ππ
οΏ½β
πΏπΏ
2ππ
πππ΅π΅,π§π§ = −ππππ ⋅ πππ΅π΅
ππππ ππππ
Elektronenspin
(Eigendrehimpuls)
Spin des Elektrons
Z-Projektion des
Spins
Magnetisches
Dipolmoment des
Spins
ππ = οΏ½π π (π π  + 1) ⋅ β
1
π π  =
2
πππ§π§ = πππ π  ⋅ β
1
πππ π  = &plusmn; οΏ½ οΏ½
2
ππβ
πππ π  = − οΏ½ οΏ½ ππβ
οΏ½οΏ½οΏ½β
ππππ
οΏ½οΏ½οΏ½β
πππ π  = −2πππ΅π΅ ππβ
11
πππ π ,π§π§ = &plusmn;πππ΅π΅
πππ΅π΅ =
οΏ½οΏ½οΏ½οΏ½β
−ππ
πΏπΏοΏ½β
2ππ
πΈπΈππβ = βππ
ππ ⋅ ππ = ππ
β ⋅ ππ β
=
ππ
ππ
β ⋅ ππ
ππππβ = 2
ππ
ππ =
πΈπΈππβ = πΎπΎ + ππππ
β ⋅ ππ = πππ π π π π π π π  ⋅ ππ + ππππ
β
ππππ
πππ π π π π π π π  = οΏ½ οΏ½ ⋅ ππ −
ππ
ππ
πππ π ,π§π§ = &plusmn;πππ΅π΅
ππ =
β
ππ
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Constants
Speed of light
Elementary charge
Electron mass
Proton mass
Boltzmann constant
Planck constant h,
β
oder β = 2ππ
Bohr magneton
ππ = 2.998 ⋅ 108 ππ⁄π π
ππ = 1.60210 ⋅ 10−19 πΆπΆ
ππππ = 9.11 ⋅ 10−31 ππππ
ππππ = 1.67 ⋅ 10−27 ππππ
ππ = 1.38 ⋅ 10−23 π½π½⁄πΎπΎ
β = 6.626 ⋅ 10−34 π½π½π½π½
β = 1.054 ⋅ 10−34 π½π½π½π½
ππ
=
⋅
πππ΅π΅ = 9.274 ⋅ 10−24 π½π½⁄ππ
πππ΄π΄
= 6.0225
⋅ 1023 ππππππ −1
12
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Version
updated
2.3
- Formatting, some minor corrections
2.4
2.5
2.6
- update: mechanics formula for air
resistance
2.7
Added: formulas for physics of atoms
Added: stern – triangle – formula
minor corrections
2.8
Minor corrections
3.0
- added: some formulas for oscillations
3.2
- added: some formulas of atom physics
- added: some formulas of electronics
22.2.18
3.3
- added: some formulas standing waves
11.6.18
4.0
- Change to english
- minor corrections
4.1