PHYSICS 221 Exam 2 Spring 2012Formula/Information Sheet

• Basic constants:

Gravitational acceleration g = 9.8 m/sec2

Permittivity of free space ε0 = 8.8542× 10−12 C2/N·m2 [ k = 1/4πε0 = 8.9875× 109 N·m2/C2]Permeability of free space µ0 = 4π × 10−7 T·m/A [ km = µ0/4π = 10−7 Wb/A·m]Elementary charge e = 1.60× 10−19 CUnit of energy: electron volt 1 eV = 1.60× 10−19 JUnit of energy: kilowatt-hour 1 kWh = 3.6× 106 JPlanck’s Constant h = 6.626× 10−34 J sec

• Properties of some particles:Particle Mass [kg] Charge [C]

Proton 1.67× 10−27 +1.60× 10−19

Electron 9.11× 10−31 −1.60× 10−19

Neutron 1.67× 10−27 0

• Some indefinite integrals: ∫dxx

= lnx∫

dxa+bx

= 1b

ln (a+ bx)∫dx

(x2+a2)3/2= x

a2√x2+a2

∫x dx

(x2+a2)3/2= − 1√

x2+a2∫dx√x2±a2

= ln (x+√x2 ± a2)

∫x dx√x2±a2

=√x2 ± a2

• Equations for Periodic Motion and Waves:

f = 1T

ω = 2πfHook’s Law Fx = −kx

ω =√

km

General solution to SHM x(t) = A cos(ωt+ φ)Total Energy in SHM E(t) = 1/2mv2

x + 1/2kx2

Torsional Pendulum ω =√

κI

Simple Pendulum ω =√

gl

Physical Pendulum ω =

√mgdI

Damped Harmonic Motion Solution x(t) = Ae−b/2m)t cos(ω′t)

ω′ =√

(k/m− b2/4m2)

A = Fmax√(k−mω2

d)2+b2ω2

d

Wave velocity v = λf

Wave Equation ∂2y(x,t)

∂x2= 1

v2∂2y(x,t)

∂t2

General solution to the wave equation y(x, t) = A cos(ω(x/v − t))y(x, t) = A cos(kx− ωt)

Velocity of a wave on a string v =√

Fµ

Average Power in a wave Pave = 1/2√µFω2A2

Inverse Square Law of Intensity I1I2

=r22r21

For strings fixed at both ends fn = n(v/2L)(n = 1, 2, 3, ..)

f1 = 12L

√Fµ

Pressure in a sound wave pmax = BkA

Longitudinal wave in a fluid v =√

Bρ

Longitudinal wave in an ideal gas v =

√γRTM

Longitudinal wave in a solid rod v =√

Yρ

Sound Intensity I = 1/2√ρBω2A2

=p2max2ρv

Definition of Intensity Levels β = (10dB) log II0

reference sound intensity I0 = 10−12W/m2

Standing sound waves, open pipe fn = nv2L

(n = 1, 2, 3..)Standing sound waves, stopped pipe fn = nv

4L(n = 1, 3, 5, ...)

Beat Frequency fbeat = fa − fbDoppler Effect fL = v+vl

v+vsfs

• Basic Equations for Waves, Interference and Diffraction:

Wave Equation ∂2f(x,t)

∂x2= 1

v2∂2f(x,t)

∂t2

Plane EM wave traveling in the +x direction E(x, t) = Em sin(kx− ωt)B(x, t) = Bm sin(kx− ωt)

Speed of an EM wave [m/s] c = 1√µ0ε0

= EmBm

= E(x,t)B(x,t)

Wave length of an EM wave [m] λ = cf

Wave number of an EM wave k = 2πλ

Poynting vector [J/s·m2] ~S = 1µ0

~E × ~B

Time-averaged S [J/s·m2] Save = EmBm2µ0

Intensity of an EM wave [J/s·m2] I = SaveTotal energy of an EM wave [J] U = I A tTotal momentum of an EM wave |~p| = U

c

Law of Reflection θincident = θreflected

Snell’s Law n1 sin(θ1) = n2 sin(θ2)Law of Malus I = Imax cos2 φBrewster’s Law tan θp = nb

na

Mirror Equation 1f

= 1s

+ 1s′

Refraction from a Spherical Surface nb−na

R= na

s+ nb

s′

Lens Equation 1f

= 1s

+ 1s′

Lens Maker’s Equation 1f

= (n− 1)(

1R1− 1

R2

)Magnification M = hi

ho= −s′

s

Double Slit Constructive Int. d sin(θ) = mλDouble Slit Destructive Int. d sin(θ) = (m+ 1

2)λ

Intensity Maxima Iθ = Io cos2(φ/2)φ = 2πd

λsin(θ)

Single Slit Diffraction Dest. Int. sin(θ) = mλa

Single Slit Diffraction Intensity I = Io

[sin(πa(sin θ)/λ)πa(sin θ)/λ

]2

Multiple Slit Diffraction Const. Int. d sin(θ) = mλ

Two Slit Diffraction Intensity I = Io cos2 φ2

[sin(β/2)β/2

]2

φ = 2πdλ

sin(θ)β = 2πa

λsin(θ)

Diffraction Grating d sin(θ) = mλResolving Power R = λ

∆λ= Nm

X-Ray Diffraction 2d sin(θ) = mλResolving Power of Circular Apertures sin(θ1) = 1.22 λ

D