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List of useful formulae for Phys 123, Midterm 3

Ch. 14

Hookes law for spring: F = !k x , k is spring constant

For a horizontal mass-spring system: E = mv2/2 + kx

2/2 , E is total energy

Simple Harmonic Motion: x(t) = A cos ( !t + ")

f = 1/T ; != 2"f ; vmax = !A ; amax = !2A

For a mass-spring system: ! = (k/m)1/2 ; for a simple pendulum: #= (g/L)1/2

Energy of an under-damped oscillator: E(t) = (#) m #2A2(t) = Eoexp[t/$]

where $ is the time constant

Quality Factor Q = #o$%2&/ (|'E|/E)cycle

Ch. 15

Speed of a string wave: vstring = (Tension/)1/2

, = Mass/Length (kg/m)

Harmonic Wave Function: y(x,t) = A sin ( kx #t + ") with k = 2&/(

Speed of a harmonic wave: v = f $ = !/k

Intensity of a wave: I = Pav/ A

Intensity level of a sound wave: )= (10 dB) log (I/I0)

For a wave incident on a boundary surface: r = (v2!v1) / (v2+v1) $= 2v2/ (v2+v1)

Doppler Effect for Sound: fr = fs ( v ur) / ( v us)where: ur= velocity of receiver , us= velocity of source , v = speed of sound

Ch. 16

sin*1+ sin*2 = 2 cos[(*1!*2)/2] sin[(*1+*2)/2]

Beat frequency: %f = f1 f2

Phase difference due to path difference: "= 2&( 'x / ()

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For standing waves on a string fixed at both ends: fn = nf1

with f1= v/2L and n = 1,2,3,

For standing waves on a string fixed at one end, free at the other: fn = nf1

with f1 = v/4L and n = 1,3,5,

Ch. 30

For a plane EM wave moving to the right along +x axis:

Ey = E0sin(kx !t) ; Bz = B0sin(kx !t) ; B0= E0/ c

c2 = 1 / (0 &0) c = 3 +10

8 m/s

Ch. 31

Index of Refraction: n = c/v with nair= 1.0

Law of Reflection: *1,= *1

Snells Law of Refraction: n1sin '1 = n2sin '2

Total Internal Reflection: sin 'C = n2/ n1 with n1> n2

Polarization by Absorption (Law of Malus): I = I0cos2'

Polarization by Reflection: n1sin*P= n2sin*2 with *2= 90o!*P

Ch. 32

Mirror equation: 1/s + 1/s, = 1/f where f = r/2

Lateral magnification by mirror: m = y,/y = !s,/s

Image formed by refraction: n1/s + n

2/s, = (n

2!n

1) / r

Magnification due to refraction: m = y,/y = !(n1s,) / (n2s)

Lens-makers equation: 1/f = (n/nair!1) (1/r1!1/r2)

Thin-Lens equation: 1/s + 1/s, = 1/f

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Power of a lens: P = 1/f (in Diopters = m-1

)

Lateral magnification by lens: m = y,/y = !s,/s

Angular magnification of a Simple Magnifier: M = */ *o= xnp/ f

Magnifying power of a Microscope: M = moMe w/ mo= !L/fo and Me= xnp/fe

Magnifying Power of a Telescope: M = *e/*o w/ tan*o= y/s = !y,/fo and tan*e= y,/fe

Ch. 33

Phase difference due to path-length difference: "= 2&('r/()

Two-slit Interference maxima: d sin'm= m $ where m = 0,1,2,3,

Two-slit Interference minima: d sin*m= (m !#) ( where m = 1,2,3,

Distance of mth

fringe from the central point of the screen: ym= L tan'm

Intensity of light from interference: I = 4I0cos2((/2)

Diffraction minima for single slit of width -: -sin'n = n $ where n = 1,2,3,

First diffraction minimum for circular aperture of diameter D: sin'1= 1.22 $/ D

Resolving power of a grating: R = (/ |'(| = mN , m = order number, N = # of slits

Ch. 34

Energy of a photon: E = hf = hc/$ and E = pc

Photoelectric effect: Kmax = hf !) ()= work function)

Compton scattering: $s!$i= $C(1 !cos') with $C = h/mec = 2.426 pm

de Broglie wavelength of a particle of mass m: $= h/p = h/(mv) = h/(2mK)#

Probability of finding a particle in a region of length dx: P(x) dx =.2(x) dx

Heisenbergs Uncertainty Principle: %x %px/#hbar with hbar= h/2"

Energies of a particle in a one-dimensional box of length L: En= n2h

2/ (8mL

2)

with n = 1,2,3, quantum number

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Standing wave functions for a particle in a one-dimensional box of length L:

.n(x) = (2/L)1/2

sin(n&x/L)

Expectation value of a function F(x): 0F(x)1=2F(x).2(x) dx

Bound state energy levels for Hydrogen: En = !13.6 eV/n2 with n = 1,2,3,

Constants and Conversion Factors

Acceleration of gravity: g = 9.81 m/s2

e = 1.6 +10 19 Coulombs

1 eV = 1.6 +10-19 J

Speed of light: c = 3 +108 m/s

Plancks constant: h = 6.63 +10-34 J3s = 4.14 +10-15 eV3s

hc = 1240 eV3nm

Powers of 10

109 giga G

106 mega M

103 kilo K10

-3 milli m

10-6 micro

10-9

nano n10

-12 pico p

10-15 femto f