integrator/differentiator - physicsphysicsweb.phy.uic.edu/481/statistics-iv.pdf · ·...

Integrator/Differentiator

October 10, 2016 Operational Amplifiers, Zhenyu Ye 1

Integrator Differentiator

Vout = 1/(RC) * int Vin dt Vout =(RC) * dVin/dt

Integrator/Differentiator

31-Oct-16 Statistics-III, Zhenyu Ye 2

𝐺"#$ =𝑉'($𝑉"#

= −

1𝑗𝜔𝐶𝑅 =

𝑗𝜔𝐶𝑅

𝐺"#$ =𝑉'($𝑉"#

= −𝑅1𝑗𝜔𝐶

= −𝑗𝜔𝐶𝑅

Statistics - IV

Zhenyu Ye


R.J.Barlow Statistics: a guide to the use of statisticalmethods in the physical sciences Chapter 5-8

ML vs LS Methodsn ML find the parameter values that maximize

likelihood function, while LS find the parametervalues that minimize 𝜒0 = ∑ 2345(73;9)

;3

0<"=>

n In a linear model, if the errors belong to a normaldistribution the Least Squares estimators are alsothe Maximum Likelihood estimators

n Uncertainty of the LS estimators can be numericallydetermined by finding parameter values which give𝜒0 = 𝜒?"#0 + 1 (equivalent to −𝑙𝑛𝐿?"# + 0.5)


Chi-squared Distributionn The chi-squared distribution (also chi-square or

χ²-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variable.


Chi-squared Distributionn The chi-squared distribution (also chi-square or

χ²-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variable.


Mean = kVariance = 2k

Goodness of Fitn The p-value is the probability of observing a test

statistic at least as extreme in a chi-squared distribution.


Goodness of Fit


Goodness of Fit


P-value ~ 0.35

Milikan’s Oil Drop Experiment




P(Chi2=53.81;NDF=7)=2.34x10-9

Advanced Labs – Milikan Oil Drop

Zhenyu Ye





𝐹H

0

yE

q: charge of the oil dropletE: electrical field strengthV: voltage applied to two parallel platesd: distance between the plates

𝐹H = 𝑞 J 𝐸 = 𝑞 J𝑉𝑑



𝐹H

𝐹M

0

yE

m: mass of the oil dropletg: gravity acceleration𝜌: density of the oil dropleta: radius of the oil droplet

𝐹H = 𝑞 J 𝐸 = 𝑞 J𝑉𝑑

𝐹M = −𝑚 J 𝑔 = −43𝜋𝜌𝑎

U J 𝑔



𝐹H = 𝑞 J 𝐸 = 𝑞 J𝑉𝑑𝐹H

𝐹M 𝐹M = −𝑚 J 𝑔 = −43𝜋𝜌𝑎

U J 𝑔

𝐹5 𝐹5 = −𝑘 J 𝜐 = −6𝜋𝜂𝑎 J 𝜐

0

yE

𝜐

k: friction coefficient between oil and air𝜐: speed of the oil droplet𝜂: viscosity of the aira: radius of the oil droplet

Stoke’s Law



𝐹H = 𝑞 J 𝐸 = 𝑞 J𝑉𝑑𝐹H

𝐹M 𝐹M = −𝑚 J 𝑔 = −43𝜋𝜌𝑎

U J 𝑔

𝐹5 𝐹5 = −𝑘 J 𝜐 = −6𝜋𝜂Z55𝑎 J 𝜐

0

yE

𝜐

𝜂Z55 = 𝜂 >>[ \

]^

when 𝜐 < 0.1 cm/s

b: constantp: air pressure



0 = 𝐹H + 𝐹M + 𝐹5

𝐹H

𝐹M

𝐹5

0

yE

𝜐

When the oil droplet reaches the terminal velocity, i.e. a=0

−43 𝜋𝜌𝑎

U J 𝑔 − 6𝜋𝑎 J𝜂

1 + 𝑏𝑝𝑎

J 𝜐b = 0

With E=0:

⇒ 𝑎 = (𝑏2𝑝)

0−9𝜂𝜐b2𝑔𝜌 −

𝑏2𝑝



𝐹H

𝐹M

𝐹5

0

yE

𝜐

𝑞 J 𝐸 −43𝜋𝜌𝑎

U𝑔 − 6𝜋𝜂

1 + 𝑏𝑝𝑎𝜐 = 0

d: distance between plates, measurable by caliper

𝜌: density of oil, 886 kg/m3

g: gravity acceleration, 9.8 m/s2

b: 8.22x10-3 Pa*mp: atmosphere pressure, 1.013x105 Pa

With E≠0:



𝐹H

𝐹M

𝐹5

0

yE

𝜐 ⇒ 𝑞 = −4𝜋𝑔𝜌 J 𝑠3𝜐b

(𝑏2𝑝)


𝑏2𝑝

U


U𝑔 − 6𝜋𝜂

1 + 𝑏𝑝𝑎𝜐 = 0

With E≠0:

Calipers

September 12, 2016 Linear Circuits, Zhenyu Ye 23

1. Outside large jaws: used to measure external diameter or width of an object2. Inside small jaws: used to measure internal diameter of an object3. Depth probe: used to measure depths of an object or a hole4. Main scale: scale marked every mm5. Main scale: scale marked in inches and fractions6. Vernier scale gives interpolated measurements to 0.1 mm or better7. Vernier scale gives interpolated measurements in fractions of an inch8. Retainer: used to block movable part to allow the easy transferring of a measurement

Calipers

September 12, 2016 Linear Circuits, Zhenyu Ye 24

1. Outside large jaws: used to measure external diameter or width of an object2. Inside small jaws: used to measure internal diameter of an object3. Depth probe: used to measure depths of an object or a hole4. Main scale: scale marked every mm5. Main scale: scale marked in inches and fractions6. Vernier scale gives interpolated measurements to 0.1 mm or better7. Vernier scale gives interpolated measurements in fractions of an inch8. Retainer: used to block movable part to allow the easy transferring of a measurement

Oil Droplet Charge


𝑞 = −4𝜋𝑔𝜌 J 𝑠3𝜐b

(𝑏2𝑝)


𝑏2𝑝

U

= −8.74×104>k C

=5.5 qe

qe=-1.6021766208(98)×10-19 C

Error Estimationn It is not the difference to the expected

value.


Error Estimation



(𝑏2𝑝)


𝑏2𝑝

U

1.65e-15 1.85e-13

Error Estimation


≈ −4𝜋𝑔𝜌 J 𝑠3𝜐b

−9𝜂𝜐b2𝑔𝜌

U/0


(𝑏2𝑝)


𝑏2𝑝

U

Error Estimation


≈ −4𝜋𝑔𝜌 J 𝑠3𝜐b

−9𝜂𝜐b2𝑔𝜌

U/0


(𝑏2𝑝)


𝑏2𝑝

U

=−noMpU − kq

0Mp

U/0J 𝑠 𝜐b

= −(8.74 ± 0.33)×104>k C

=(5.5±0.2) qe qe=-1.6021766208(98)×10-19 C

Advanced Labs - Zeeman Effects

Zhenyu Ye



Modeling of Hydrogen Atoms

October 3, 2016 Advanced Lab II, Zhenyu Ye 31

n Schrodinger equation in 1926

i! ∂∂tΨ!r, t( ) = −!2

2m∇2 +V !r, t( )

⎡

⎣⎢

⎤

⎦⎥⋅Ψ

!r, t( )

Ψ!r( ) = 1

r⋅ χ l r( ) ⋅Ylm θ,φ( )

En = −e2

!c⎛

⎝⎜

⎞

⎠⎟

2mec

2

2n2

m = 0,±1,!,±l

L = l(l +1)! Lz =m!

l = 0,1,!,n−1n =1,2,!

Electron Spin

11/7/16 Advanced Lab IV, Zhenyu Ye 32

S = s(s+1)! s = 12 Ag Shell Structure:

2, 8, 18, 18, 1

1925: G.Uhlenbeck, S.Goudsmit

µs = gsmsµB

Sz =ms! ms = ±12

Stern-Gerlach Experiment 1922

En,ml ,ms= −

e2

!c⎛

⎝⎜

⎞

⎠⎟

2mec

2

2n2+ glmlµBB+ gsmsµBB

𝜇t =𝑒ℏ2𝑚

𝑔w = 1

𝑔x = 2

L, S and J


2S+1LJ

541.6nm

S=1, L=0, J=1

S=1, L=1, J=2

L, S and J


𝑔y =𝑔x J 𝑆 + 𝑔w J 𝐿

𝑆 + 𝐿

2S+1LJ

𝐸 = 𝐸t=b + 𝑔y𝑚y𝜇t𝐵

∆𝐸 = ∆(𝑔y𝑚y)𝜇t𝐵

541.6nm

L, S and J


ΔJ=±1, Δmj=0, ±1

𝑔y =𝑔x J 𝑆 + 𝑔w J 𝐿

𝑆 + 𝐿

2S+1LJ

𝐸 = 𝐸t=b + 𝑔y𝑚y𝜇t𝐵

∆𝐸 = ∆(𝑔y𝑚y)𝜇t𝐵

541.6nm

Zeeman Effect Lab


Polarizer


Interference Filter


=𝜆4

Fabry-Perot Etalon


𝑘𝜆 = 2𝑑𝑐𝑜𝑠𝜃 = 2𝑑 1 − 𝑠𝑖𝑛0𝜃 ≈ 2𝑑 1−𝜃0

2 ≈ 2𝑑 1−𝐷�0

8𝑓0

Fabry-Perot Etalon


∆𝜆 =𝜆0

2𝑑𝐷�U0 − 𝐷�00

𝐷�4>0 − 𝐷�00=𝜆0

2𝑑𝐷�00 − 𝐷�>0

𝐷�4>0 − 𝐷�00

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