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PHYS1001 Equation Sheet

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The following equations and data might be useful:
Acceleration due to gravity at earths surface:
Speed of light in a vacuum
Permittivity of free space
Mass of one proton
Mass of one electron
Charge of an electron
Boltzmann’s constant
Planck’s constant
g = 9.80 m s−2
c = 3.00 × 108 m s−1
0 = 8.85 × 10−12 F m−1
mp = 1.673 × 10−27 kg
me = 9.11 × 10−31 kg
e = 1.60 × 10−19 C
kB = 1.38 × 10−23 J K−1
h = 6.626 × 10−34 J s
= 4.136 × 10−15 eV s
h
~=
= 1.055 × 10−34 J s
2π
= 6.582 × 10−16 eV s
The Stefan-Boltzmann constant
Rydberg
σ = 5.67 × 10−8 W m−2 K−4
R∞ = 13.606 eV
The universal gas constant
R = 8.314 J K−1 mol−1
Specific heat of liquid water
Specific heat of ice
cwater = 4184 J kg−1 K−1
cice = 2050 J kg−1 K−1
Heat of fusion for water
Lf,water = 3.34 × 105 J kg−1
Heat of vapourisation for water
Speed of sound in air
Lv,water = 2.26 × 106 J kg−1
= 330 m s−1
Density of water (20◦ C and 1 atm)
Conversion factors:
= 1.00 × 103 kg m−3
1 eV = 1.60 × 10−19 J
0 ◦ C = 273.15 K
1 L = 10−3 m3
Prefixes:
f = 10−15 , p = 10−12 , n = 10−9 , µ = 10−6 , m = 10−3 , k = 103 , M = 106 , G = 109 , T = 1012
Heat & Thermodynamics equations
Thermal expansion:
∆L = αLi ∆T
Heating/Cooling:
Q = mc∆T
Thermal motion in a gas:
2 = 3 kT
Kave,trans = 12 mvth
2
Heat Transfer by radiation:
4 − T 4)
Pnet = Pabs − Pem = eσA(Tenv
Heat Transfer by conduction:
H=
First Law and Work:
∆U = Q + W
Ideal Gas Law:
pV = nRT
Internal energy (ideal gas):
U = 32 nRT (monatomic)
γ (ideal gas):
γ=
Q
t
∆V = β Vi ∆T
β = 3α
Q = mL
−Tc
= kA Th L
vth =
p
3kT /m
H is heat flow in watts.
RV
W = − Vif p dV
Where n is the number of moles of gas.
∆U = nCV ∆T
Cp
CV
γmonatomic =
γdiatomic =
5
3
7
5
γpolyatomic =
4
3
Vf
Vi
Work (ideal gas):
Wisothermal = −nRT ln
Specific Heat (ideal gas):
Q = nCV ∆T
Q = nCp ∆T
Cp = CV + R
CV = f2 R
Adiabatic process in ideal gas:
pV γ = constant
Entropy change:
∆S =
Q
T
Wadiabatic =
T V γ−1 = constant
Rf
(constant T )
∆S = i dQ
T
Page 2
pf Vf −pi Vi
γ−1
∆S = mc ln
Tf
Ti
Mechanics equations
p = mv
v =ω×r
dp
dt
τ =r×F
F=
L = Iω
τ =
L=r×p
Iz =
dL
dt
X
mi ri2
i
1
Ktrans = mv 2
2
dU
F (x) = −
dx
Circular motion:
Krot
1
= Iω 2
2
f ≤ µs N
Fgrav
GM m
=−
r2
f = µk N
GM m
Ugrav = −
+ U0
r
Z x
F (x)dx = U (x0 ) − U (x)
x0
dr
v=
= r ϑ̇ êϑ
dt
dv
a=
= −r ϑ̇2 êr = −ω 2 r
dt
Page 3
Waves & Optics equations
THIS SECTION WILL BE FILLED IN LONG BEFORE YOUR FIRST WAVES &
OPTICS TEST
Page 4
Electricity equations
Vsphere = 43 πr3
Asphere = 4πr2
Acircle = πr2
Ccircle = 2πr
Electric Force:
~
F~ = q E
Electrical Potential:
V =
Potential difference:
∆V = −
Field of point charge:
~ =
E
Potential of point charge:
V =
Gauss’s Law:
ΦE =
Electric Field from Potential:
~ = −∇V
~ = −( ∂V , ∂V , ∂V )
E
∂x ∂y ∂z
Field of Sphere:
~ =
E
Field near conducting sheet:
Esheet =
σ
0
Field near non-conducting sheet:
Esheet =
σ
20
Field of linear charge distribution:
Eline =
Capacitance:
C=
Parallel plate Capacitor:
Ck =
U
q
Rb
a
1 q
4π0 r2
1 q
4π0 r
~ · d~l
E
r̂
~ · dA
~=
E
H
1 q
4π0 r2
r̂
λ
2π0 r
Q
V
0 A
d
∆V = Ed
Capacitor Energy:
UC = 12 CV 2
Page 5
qenclosed
0
Breakdown of Classical Physics equations
E
c = f λ,
f= ,
h
p
2
2
p| c + m2 c4 ,
E = |~
λ=
h
p
I(T ) = σT 4
λmax T = 2.898 × 10−3 m K
λ
sin θ = m , m = 1, 2, 3, . . .
a
h
L = n~ = n , n = 1, 2, 3, . . .
2π
E = |~
p|c for photons
Kmax = hf − Φ,
h
(1 − cos φ)
mc
p
2m(U − E)
−2bL
T ≈e
, where b =
~
2
2
n 0 h
rn =
,
me vr = n~
Zπe2 me
∆λ = λ0 − λ =
∆p ∆x ≥ ~,
E=−
Page 6
∆E ∆t ≥ ~
RZ 2
, where R = 13.606 eV
n2
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