Useful Constants
R = 8.31451J/mol K, 1 atm = 1.013 × 105 Pa, 1 atm litre = 101.3 J
One can use 1 atm ≈ 1 × 105 Pa and 1 atm litre ≈ 100 J. StefanBoltzmann constant σ = 5.6704 ×
10−8 W m−2 K −4 .
γair = 1.4, CV air = 20.8Jmol−1 K −1 , The density of water is 1 gram/cm3 =1000 kg/m3 .
Mechanics
Linear Motion: x = x0 + 21 (vx0 + vx )t, x = x0 + vx0 t + 21 ax t2 , vx = vx0 + ax t,
vx 2 = vx0 2 + 2ax (x − x0 )
2
Circular Motion: ac = vr
d
Forces: F~ = m~a, F~ = dt
p~, Friction: F = µN , Spring: F = −kx, Damping: F~ = −b~v
Bouyant: FB = ρV g
R ~r
~ v.
W = ~rif F~ · d~r, W = F~ · ∆~r, K = 21 mv 2 , ∆U gravity = mgy, ∆U spring = 12 kx2 , P = dW
dt , P = F · ~
Thermodynamics
∆L
Thermal Expansion: ∆L = αL∆T Stress and Strain: F
A =Y L
3
1
Ideal Gas Law: P V = nRT , Kav = 2 kT , 2 kT for each degree of freedom.
∆T
Thermal Conductivity: I = ∆Q
∆t = kA ∆x
Black Body Radiation: P = eσAT 4 , λmax T = 2.8977685 × 10−3 m · K
Internal Energy: U = nCV T
First Law: dQ = dU + dW for an ideal gas dW = P dV .
Work for isothermal process W = nRT ln(Vf /Vi ).
For adiabatic expansion T V γ−1 = constant, if the number of moles is constant P V γ = C where C is a
constant and γ = Cp /Cv .
Work for adiabatic process
Z
V2
W =
Z
V2
P dV = C
V1
V1
dV
C
=
(V2 1−γ − V1 1−γ )
γ
V
1−γ
Q = mc∆T , Q = mL. CP = CV + R, CV = f2 R where f =degrees of freedom. f = 3 for monatomic and
f = 5 for diatomic.
dS = dQ/T
e = W/QH COPCooling = |QC |/|W | COPHeating = |QH |/|W |
eCarnot = 1 − TC /TH
Integrals
Z
xn dx =
xn+1
+ constant
n+1
Z
n 6= −1
x−1 dx = ln x + constant
Trig
sin θ1 + sin θ2 = 2 cos(
θ1 − θ2
θ1 + θ2
) sin(
)
2
2
Area and Volume Area of a sphere A = 4πr2 . Area of a cylinder A = 2πrl. Area of a circle A = πr2 .
Volume of a cylinder V = lπr2 . Volume of a sphere V = 34 πr3 .
Oscillations
ω = 2πf , T = f1 , x = Acos(ωt + φ), ω 2 =
bt
k
m.
Damped Oscillations: x = A0 e− 2m cos(ωt + φ), where ω =
q
b 2
E
ω0 2 − ( 2m
) , Q = 2π ∆E
.
bt
−m
Energy for damped E = E0 e
Waves
q
1
2 2
v = Tµ , k = 2π
λ , v = λf , P = 2 µω A v, p0 = ρωvs0
q
v
(1± vD )
Pav
I
0
v = γRT
M , I = 4πr 2 , β = 10dB log10 ( I0 ) Doppler Effect f = f0 (1∓ vs )
v
y = A cos(kx ∓ ωt + φ)
Interference k∆x + ∆φ = 2πn or π(2n + 1) n = 0, ±1, ±2, ±3, ±4, . . .
mv
Standing Waves fm = mv
2L m = 1, 2, 3, . . . fm = 4L m = 1, 3, 5, . . .
Beats ∆f = f2 − f1
2
1
Constants: k = 4π
≈ 9 × 109 N m2 /C 2 , 0 = 8.84 × 10−12 NCm2 , e = −1.6 × 10−19 C
0
µ0 = 4π × 10−7 T m/A, c = √10 µ0 = 299, 792, 458 m/s
kq
~ = k|q|
Point charge: |F~ | = k|qr12q2 | , |E|
r 2 , V = r + Constant,
Rb
R
~ · d~l = − a E
~ · d~l. Also Ex = − dV and
Electric potential and potential energy: ∆V = Va − Vb = a E
dx
b
~ = −∇V
~ .
E
∆U = Ua − Ub = q(Va − Vb )
Maxwell’s Equations:
Z
Z
Qenc
~
~
~ · dA
~=0
E · dA =
= 4πkQenc
B
0
S
S
Z
Z
~ · d~l = − dΦB
~ · d~l = µ0 Ienclosed + 0 dΦE
E
B
dt
dt
C
C
R
R
~ · dA
~ and ΦB = B
~ · dA.
~
where S is a closed surface and C is closed curve. ΦE = E
Energy Density: uE = 12 0 E 2 and uB =
1
2
2µ0 B
(energy per volume).
~ + q~v × B,
~ F~ = I L
~ × B.
~
Forces: F~ = q E
2
Capacitors: q = CV, UC = 12 qC ,
A
For parallel plate capacitor with vacuum (air): C = 0d , Cdielectric = KCvacuum
1
2
Inductors: EL = −L dI
dt , UL = 2 LI , where L = N ΦB /I and N is the number of turns.
F or a solenoid B = µ0 nI where n is the Number of Turns per unit length.
DC circuits: VR = IR, P = V I, P = I 2 R
(For RC circuits) q = ae−t/τ + b, τ = RC, a and b are constants.
(For LR circuits) I = ae−t/τ + b, τ = L/R, a and b are constants.
AC circuits: XL = ωL
q XC = 1/ωC VC m = XC Im , VL m = XL Im
2
Im
2
R, Irms = √
Vm = ZIm , Z = (XL − XC ) + R2 Paverage = Irms
2
If V (t) = Vm cos(ωt), then I(t) = Im cos(ωt − φ), where tan φ =
~ =
Additional Equations: dB
Rt
XL − XC
.
R
µ0 Id~l×~
r
4π r 3
LRC Oscillations: q = A0 e− 2L cos(ωt + φ), where ω =
q
R 2
) and ω0 2 =
ω0 2 − ( 2L
1
LC