Plank's Constant experiment

Purpose:


To determine experimentally the value of Plank's constant.

Method:

Use a diffraction grating, 2 meter sticks, and a voltmeter to record the wavelength and potential drop across a collection of differently colored LEDs.

The data from our experiment can be found here.
Four LEDs were chosen;
blue
red
yellow
and green
their differences in wavelength should allow for the construction of a graph of photon energy vs. wavelength.


Quantum physics predicts that photons which are emitted from an atom are created when an electron travels from an excited state to a relaxed state. The energy levels of each of those states are of definite energy, and thus every photon emitted should have an identical wavelength. If the atom which emits the electron is bound in a crystal lattice, the energy levels of each state cannot overlap with nearby atoms because of the Pauli Principle. Thus the energy levels of each atom exist in a band approximately centered around the original energy level. This allows for photons of various wavelengths to be emitted from a sample of uniform atomic nature.

The observation associated with this effect is a smearing of the observed emission line when viewed through a diffraction grating.

A light emitting diode, or LED, is an electronic device which takes advantage of this property to generate photons in the visible spectrum. It is usually fabricated using p-type and n-type silicon in such a way that the energy an electron gains by traveling across the LED is equal to the energy of the desired photon.

Upon analysis of of data, a graph of photon energy vs. frequency was obtained from Microsoft Excel.

Unfortunately, the units on the vertical axis are too small to be properly displayed, but they can be viewed in the associated spreadsheet. The slope of this graph is found to be 8.8269 E-34 J/m, which is in error of 33% when compared to the accepted value of plank's constant, 6.626 E-34.

This value of error is unsatisfactory, so a separate method was used to estimate Plank's constant.

If the data points for energy are divided by their corresponding frequency, a set of ratios is obtained.

                   E/f
Red:           6.38E-34 J/m
Yellow:      5.97E-34 J/m
Green:       7.08E-34 J/m
Blue:          6.81E-34 J/m

These numbers are much closer to the accepted value of Plank's constant, and when averaged, yields a ratio of 6.56E-34 J/m, which is in error by only a single percent.

Wavelength of a gas lamp

Quantization of energy levels in atomic nuclei results in different colors of emission spectra for different elements. For example, this picture features a mecury gas lamp viewed through a diffraction grating.

It is clear that there are three spectral images formed to the right of the actual lamp. These three spectral images have colors corresponding to the wavelengths of photons emitted from mercury when one of its electrons is excited and then relaxes to a lower energy state.

Our experimental apparatus, as visible in the picture, consists of a 2 meter stick, and a meter stick arranged orthogonally on a table. The horizontal displacement of the spectral images can thus be converted into an angular relationship, or it can be left in units of length to solve for the wavelength of the light directly.

Similar to the CD diffraction experiment, the position of the spectral images and the spacing of the diffraction grating can be used to determine wavelength. The governing equation is:

λ = (D*d)/sqrt(L^2+D^2)
Where:
D = the horizontal distance of the image
d = the diffraction spacing
L = the distance between the diffraction grating and the gas lamp

This experimental setup yielded the following data for a hydrogen lamp placed in the same location:

position of image in cm:
1=48
2=54
3=74.5
uncertainty: .5cm


L = 2 meters
d = .000002 meters

This setup yields the following prediction for the wavelengths

1 = 466nm

2 = 523nm
3 = 698nm

Calibration of our apparatus against a white light, yielded a correction factor for our setup which included a systematic error of +37.7nm

When factored into the previous data, our final results are

1 = 428.3nm
2 = 485.3nm
3 = 660.3nm

when compared to the actual spectra for hydrogen:

1 = 434 nm
2 = 486 nm
3 = 656 nm

It appears our predictions are correct to within a single percent. This falls approximately in the same range as our uncertainty in horizontal position.

Particle in a box

Welcome to another edition of SOLVE THAT ACTIVPHYSICS PROBLEM. Our contestant today must answer 10 questions related to the particle in a box simulation at:
http://wps.aw.com/aw_young_physics_11/13/3510/898597.cw/index.html

Lets get this contest started.
Questions in order:
Question 1: Standing Waves
From your study of mechanical waves, what is the longest wavelength standing wave on a string of length L?

The longest wavelength for a standing wave is twice the length of the string, or twice the distance between the two hard boundaries.

Question 2: The de Broglie Relation
What is the momentum of the longest wavelength standing wave in a box of length L?

 de Broglie hypothesized that the momentum of a particle must be equal to plank's constant divided by the wavelength, which in this case is 2L.

Question 3: Ground State Energy
Assuming the particle is not traveling at relativistic speeds, determine an expression for the ground state energy. 

Ground state energy here refers to the kinetic energy stored in the particle, if we are operating at non-relativistic speeds then:
p=mv
K=.5mv^2
p^2 = (mv)^2
p^2/(2m) = K

then, applying p = h/2L
K = (h/2L)^2/(2m)
K = h^2/(8mL)
Question 4: Increasing L
If the size of the box is increased, will the ground state energy increase or decrease?
It will decrease, as the relationship places L as the divisor of the energy equation.


Question 5: The Correspondence Principle: Large Size
In the limit of a very large box, what will happen to the ground state energy and the spacing between allowed energy levels? Can this result explain why quantum effects are not noticable in everyday, macroscopic situations?

As L approaches the scale of human objects, the ground state energy takes on a magnitude of 10^-37 Joules, which is negligible on macroscopic scale.

Question 6: The Correspondence Principle: Large Mass
In the limit of a very massive particle, what will happen to the ground state energy and the spacing between allowed energy levels?

Massive particles also act as divisors in the energy equation, thus large particles have low levels of ground stage energy.


Question 7: Ground State Probability
If a measurement is made of the particle's position while in the ground state, at what position is it most likely to be detected?

The particle is most likely to be found in the middle of the trap.

Question 8: Probability: Dependence on Mass and Size
The most likely position to detect the particle, when it is in the ground state, is in the center of the box. Does this observation depend on either the mass of the particle or the size of the box?

No, the wave function of the particle is independent of mass, and always has a maximum at L/2.


Question 9: Probability: Dependence on Energy Level
The most likely position to detect the particle, when it is in the ground state, is in the center of the box. Does this observation hold true at higher energy levels?

No, for instance the first excited state is more likely to be found at L/4 and 3L/4.

Question 10: The Correspondence Principle: Large n
In the limit of large n, what will happen to the spacing between regions of high and low probability of detection? Does this agree with what is observed in everyday, macroscopic situations?

As n approaches large values, the probability density function becomes almost continuously flat, or equally likely in every location. This is consistent with a free particle, which has no definite location.

Experiment 12: CD diffraction

Using the properties of diffraction, it is possible to measure the space between adjacent grooves on a CD. If you shine a laser on a CD and observe the spacing between the central maxima and the resulting accessory maxima it is possible to calculate the groove spacing by using the following geometric relationship.

Photo credits to sciwebhop

the distance between points A and B is notated as d. Θ is measured in degrees from the normal to the diffraction grating, and λ is the wavelength of the light incident on the diffraction grating. The following equation relates these variables.

d*sin(Θ)= m * λ

 Where m is the number of the maxima counting away from the central maxima.

When a laser is shone upon a CD the reflected pattern may appear something like this. 

It is clear from this image that some form of diffraction is taking place. By measuring the spacing between these maxima, and measuring the distance from the wall to our CD we can determine the angular separation of these maxima, additionally we know the wavelength of the laser to be 630-680 nanometers.

This table holds all of the relevant data.

 table

It is clear from the calculations that there is some sort of systematic error in our data collection, and it most likely stems from the fact that we were not measuring the appropriate phenomena. 

The predicted groove spacing based on our data is 2.6 * 10^-5 meters which is 16 times larger than the specifications listed for a CD(1.6 micrometers).

Unfortunately, we were unable to collect data after determining the flaw in our experiment and are unable to verify the CD groove spacing with the current data.

Additionally, here is an interesting video on another method for determining the groove spacing.

http://www.youtube.com/watch?v=IKKFPtcaZpQ

Time dialation

    Einstein based the special theory of relativity on two postulates.
The first being that the laws of physics are the same in any inertial frame.
The second being that the speed of light is measured to be the same in any inertial frame. Take these two light clocks for example...

The guy standing next to the stationary clock measures the speed of light to be c.

A guy riding on the moving clock measures the speed of light to be c.

On the surface, these simple postulates seem both logical and convenient, but they lead to a plethora of paradoxes. Take for example a person in a spaceship travelling at half the speed of light, if they turn on the headlights for their spaceship, the light travels away from them at the speed of light. 

But how does this appear to a stationary observer watching the same ship?

Classical Newtonian physics predicts that if light has a velocity of c compared to the spaceship, and the spaceship has velocity .5c compared to the observer, then the light must be travelling at  1.5c compared to the stationary observer.This however violates the second postulate, that the speed of light is measured to be the same by the people in the spaceship AND the stationary observer. How is this possible?

In a deft side step, Albert invents this half baked theory of time dilation.

What is time dilation you say?

In the simple form it means that time is travelling slower for the people on the spaceship than it is for the stationary observer. (A more through approach reveals that this too produces a paradox, but lets just wave our hands and say 'OKAY Albert, what ever you say must be true.')

Well, if time is travelling at different rates for different people, how do we compare them to one another? 

BAM! We have this convenient formula.

The t' represents the magnitude of a unit of time, lets call it a second, compared to the stationary observer's time, t. The denominator has terms v, for the velocity of the travelling thing, and c for the speed of light. The denominator can also be represented by multiplying t by a term we refer to as gamma. Interesting to note is that in the limit where v approaches c, t' approaches infinity. As such, it appears all things happen simultaneously to things travelling at the speed of light.

One might ask, how do we use this? Well, the nice people over at activphysics

provided us with an example using the light clocks from above. If told that the round trip time for the moving clock is 7.45 us, measured from the stationary frame, and that the stationary clock takes 6.67 us, we can determine the value that the coefficient of t must take in the above equation. The value that expression takes is also known as the Lorentz Factor. 

A simple division results that the Lorentz factor for the moving clock is 1.117, meaning that a single second in the stationary frame lasts 1.117 seconds in the moving frame. Solving this expression for velocity results in a v of, .445c.

Measuring a Human Hair

Interference is a property of wave interaction. The presence of interference in the propagation of light is a strong argument for it being a wave.

Using a laser, it is possible to observe interference when it is projected past a human hair. A picture of the experimental set up follows.

The note card in the picture has had a hole punched in it, and a human hair is suspended across the hole and held in place by tape. A laser is aimed past the hair so that the beam is bisected by it. On the recieving surface, it is possible to observe interference effects, though this photo did not capture them. This is due to the low intensity of the constructive interference bands.

By measuring the distance from the notecard to the flat surface, the distance from the center of the laser beam to the center of one of the intensity peaks, and using the wavelength of the laser, it is possible to measure the diameter of the hair suspended in the notecard.

Our experiment generated the following results

Length from notecard to board : 2 meters
distance from laser center to 3rd anti-node (Ym): 6.5 cm

wavelength of light emitted: 630 - 680 nanometers

using the constructive interference approximation equation
Ym = Length * wavelength * anti-node number / diameter of hair

Solving this equation for diameter results in a range of diameters from 5.85*10^-5 meters to 6.27*10^-5 meters. Upon further investigation, a microscope determined that the diameter of the hair was approximately 6*10^-5 meters which supported the laser measurement.

Thin Lens experiment

As always, the data for this experiment can be found on google documents by following this link.

A magnifying lens can be modelled as a thin lens in the physics of optics because its total thickness is negligible in comparison to the radius of curvature for each surface.

For this experiment we used a light box, a paper with a non symmetric cross printed on it, a magnifying lens, and a flat surface to project the image on.

The above image shows the light box and lens combination.
When the ambient lighting is suppressed and the light box is projected through the lens, an inverted image can be seen on the opposite side of the lens.

Though barely visible in the glass, it is possible to see that the reflection of the cross is both vertically and horizontally inverted from the image that appears on the flat box.

The sharpness and size of the projected image was found to be variant with light box distance (object distance) and projected surface distance (image distance). Both measurements are in reference to the center of the lens.

In order to collect data, the light box was placed at a known distance, and the flat surface was moved until a sharp image was formed. The two relative distances were recorded and put into a table. The plot of Image distance vs. object distance follows.

It appears to have a relationship which approximates a reciprocal, however, a plot of their inverses, when summed provides a more thorough explanation.

It is apparent from this graph that the summation of the reciprocal of the two values is nearly constant. Further investigation reveals that the average value for this summation is approximately 1/focal length.

Geometric approach to optics

In the study of electromagnetic wave propagation, a simplification to geometric rays can aid understanding.
For example, when dealing with the subject of the appearance of an object when viewed in a convex mirror can be simplified to the 2 dimensional case, as it appears in the following image.

The trajectory of 3 rays originating at the top of the object is illustrated. The point at which those three rays converge, is the reflected image location. In this case, the image is smaller and inverted. It can be shown with additional rays that the reflected image acts as its own point source and is thus a real image.

Experiment # 7: Introduction to Reflection and Refraction

Using a light box and a half cylinder of acrylic, it is possible to observe light's behaviour when it changes mediums.

 

In this experiment, the acrylic object is rested on top of a paper protractor. Then the incident and refracted angles are recorded.

As always, the data associated with this experiment can be found in my Google documents folder here. The uncertainty associated with each angle measurement is plus or minus a half degree.

From the data, it appears that there are two observable trends as illustrated by the following graphs.

These graphs represent a plot of the refracted angle (Theta 2) vs. the angle of incidence (Theta 1). The graph which is titled air to acrylic represents the data collected when the light hits the flat part of the acrylic first, and the chart titled acrylic to air represents when the light hits the rounded side of the acrylic first.

First thing of note, both of these graphs have a relatively linear trendline, as demonstrated by their regression numbers(R^2 on the graph). Unfortunately, they both have non-zero intercepts with the vertical axis. Also, both appear to have some curvature in their data points as the angle increases.  This leads me to believe that the linear relationship between the refracted angle and the incident angle only applies when the small angle approximation can be used. (angles less than 30 degrees)

Additionally, it is interesting to note that the slopes of both graphs are nearly the inverse of each other.v(within 10 percent)

When the Sine of the angles are plotted against each other, a clearer picture begins to form.

 

 This relationship has a demonstrably more linear relationship, because the regression numbers are much closer to unity than the straight angle to angle relationship. Additionally, the intercepts with the vertical axis are almost zero, which supports the experimental data. (In both data sets, the point 0,0 appears).

Also, the slopes of these lines are closer to the inverse of each other.(2 percent deviation)

In titling these charts, it is important to note that the description of the interface applies to when the light contacts the linear part of the acrylic disk. When the light enters or exits the acrylic disk through the rounded portion, no refraction occurs because it always has an angle of incidence of zero. This is assuming that the light enters the disk aimed at the mid point of the linear portion of the disk, which is the case for the entirety of both data sets.

Observations:

The slope of the graph was greater when the light travelled from the acrylic to the air. Additionally, the slopes of each graph are very nearly the inverse of one another. This leads me to believe that there is some constant which represents a ratio between the sine of the angle of incidence and the sine of the angle of refraction.

According to Sears and Zemansky's University physics, the relationship between the angle of incidence and the refracted angle does depend on the sine of both angles, and is modified by 2 constants. It takes the following form:

where v1 is the velocity of light in medium 1 v2 is the velocity of light in medium 2 and n1 and n2 are some constants which describe the refractive property of a given medium.

Air is said to have a refractive index of nearly 1, so it is apparent that the acrylic disk used for the experiment has a refractive index of 1.466. Furthermore, this measurement is consistent with reported indexes of refraction for acrylic glass which fall in the range of 1.490 - 1.492. (error =1.6-1.7%)

TEST OF MATLAB PUBLISHING

Contents

• LOADING THE RELEVANT DATA
• PLOTTING THE RAW
• PLOTTING THE MEAN
• PLOTTING THE RUNNING AVERAGE

%practice script from the matlab interactive tutorial
%loading data into the matlab editor

LOADING THE RELEVANT DATA

load annual_temps
%LOADING THE RELEVANT DATA

PLOTTING THE RAW

plot(year,annual)
title('HadCRUT3 Temperature Anomaly Measurements')
xlabel('Year')
ylabel('Temperature Anomaly (°c)')
% PLOTTING THE RAW

PLOTTING THE MEAN

annual_avg = mean(annual);
figure
area(year,annual_avg)
title('Average Annual Temperature Anomaly')
xlabel('Year')
ylabel('Temperature Anomaly (°c)')

 

PLOTTING THE RUNNING AVERAGE

k = 1/5*ones(1,5);
five_year_avg = convn(annual_avg,k,'valid');
plot(year,annual_avg,'--o')
hold on
plot(year(3:end-2),five_year_avg,'r')
legend('Annual Avg', 'Five Year Avg')

Published with MATLAB® 7.13

Wave length, frequency, tension lab

Data for all of the following graphs can be found here.


In the previous experiment, a relationship was found between wavelength and period. In the following experiment, we establish that this relationship also applies to frequency, and there is a relationship between these variables which depends on the tension in the vibrating medium.

The experimental apparatus can be observed in the following video.

The set up consists of a mechanical oscillator, a function generator, a length of string, a pulley and a weight.

The string is attached on one end to the mechanical oscillator, and across the pulley to the weight on the other end. The weight serves to maintain a nearly constant tension for the duration of the experiment.

The function generator is set to generate sinusoidal voltages of varying frequency. This frequency is adjusted until a given number of anti-nodes is established in the pattern of oscillation.

For a given tension, at least 6 different standing waves were tested by varying the frequency, these standing waves represent a specific wave length as measured by comparing the space between nodes to a meter stick. A table with the relevant data is available at the top of this post.

Observations: A plot of the frequency vs. wavelength results in a plot which resembles functions of the form;
frequency = some constant divided by wavelength.

when plotted against the reciprocal of wavelength, a linear relationship is observed.

when a linear regression is added to the chart, it is evident that in order for the units in the function to make sense, the coefficient of x must have units of meters per second. (owing to the fact that 1/wavelength has units of meters, and frequency has units of 1/seconds.)

Thus, the coefficients from the linear regression can be converted to velocities. It is important to note that the units from the above graph are in cm not meters.
These velocities can be plotted against the corresponding tensions in order to establish a trend. 

In the above graph, it is evident that the blue points, (the plot of velocity vs. tension) very nearly adhere to the function f(x)=30.47x^.48.

The equation which models the velocity of a wave through a medium is:

where v is the longitudinal(parallel to the medium) velocity of the wave, and mu is the linear density of the medium. The linear density of the string was determined to be .00119 kg/m.

Also visible in the graph is a plot of the theoretical velocities which correspond to the given tensions. The function which most closely matches those values is f(x)=28.99x^.50.

The equations are similar in that the coefficient of x differs by approximately 5 percent, and the power of x differs by approximately 4 percent. This indicates a strong relationship between the measured data and the theoretical model.

A comparison of the ratio between the real velocities and the predicted velocities can be observed in the aforementioned spreadsheet.

For each step of the harmonics measured, it was observed that the required frequency was always nearly a whole multiple of the first measured frequency. (15,30,45,60...) This indicates that once the first harmonic of the apparatus is discovered, finding the requisite frequencies for any other harmonic is as simple as multiplying the first frequency by the number of the harmonic desired.

wavelength lab

The fundamental relationship between wavelength and period can be studied with a length of spring and a stopwatch.

This movie shows a standing wave set up in a spring. The benefits of using a standing wave to study wavelength are that the distance from person to person is a good approximation for the wavelength, and frequency measurements can be simplified.

For this experiment, our procedure is as follows:

• two students stand approximately 4 meters apart
• each student takes one end of a 4 meter spring and allows it to hang between them
• one student sets up an oscillation in the spring until a given number of anti-nodes are established
• a stopwatch is used to record the time for the standing wave to oscillate ten times
• the time is divided by ten in order to determine the period of one oscillation

The relationship between period on wavelength was determined for 4 data points.
all wavelengths are accurate to plus or minus .1 meter
all times are accurate to plus or minus .15 seconds
wavelength              period
8 meters                  1.04 seconds
4 meters                  .587 seconds
2.66 meters             .350 seconds
2 meters                  .262 seconds

these data points, when plotted, imply a linear relationship between wavelength and period.

This supports the formula for a standing wave, where the velocity of a wave in a given medium is a constant based on its linear density and the tension in that medium, and that velocity is equal to the wavelength divided by the period. This relationship is of the form;

Put a Hole in the Bucket Dear Liza...

Experiment 2: Fluid Dynamics

The equation which models the behaviour of a laminar, non-viscous, incompressible fluid flow is as follows:

This equation can be used to determine the theoretical velocity of water exiting a hole in a bucket.

For this experiment, we measured the rate at which water flowed out of a bucket, by recording the amount of time it took to fill a 1000 mL beaker. Data follows:

Bucket Diameter = 25 cm
Initial height of Fluid (above hole) = 12.4 cm
Area of drain hole = .31669 cm^2
Volume emptied = 1000 mL

Trial 1 = 30.01 s
Trial 2 = 29.35 s
Trial 3 = 29.59 s
Trial 4 = 29.39 s
Trial 5 = 29.82 s
Trial 6 = 29.38 s

This data indicates that the time to empty 1000 mL of water should be 29.59 ± .25 seconds. This equates to a flow velocity of:
1000cm^3/.31669cm^2=3157.66cm
3157.66cm/29.59 s=1.06m/s

The predicted velocity, from Bernoulli's equation should be close to the square root of 2*g*height of the water.
that results in a theoretical velocity of:
1.559 m/s

This velocity predicts that the volume of water should be emptied in 20.24 seconds, which represents nearly a 33% deviation from the measured time.
Even if the height of the water is adjusted to reflect the average height over the duration of the experiment, the theoretical time to completion only changes by 1.25 seconds.

These findings lead me to believe that some part of our model is incorrect, for example, the size of the drilled hole may not be exact.
The given data predicts that the cross sectional area of the hole may be in error by as much as 30 percent. This could be caused by the plastic deforming during the drilling process, and leaving a lip which blocked some of the flow.

Fluid Statics Laboratory

Re-purposing a digital force sensor as a scale is pretty efficient.

This picture shows a Vernier force sensor, attached to a laptop which is running logger pro. An aluminum cylinder is connected to the force sensor by a string.
The cylinder was measured to weigh .229 ± .006 Newtons when suspended in air.
When suspended in water, it was measured to weigh .165± .006 Newtons.
The difference between these two measurements is the buoyant force on the cylinder due to the volume of water it displaces.
The buoyant force was determined to be .064±.012 Newtons.

Another Method for determining the buoyant force is to submerge the cylinder in a vessel which is nearly overflowing. The action of submerging the cylinder will displace a volume of water from the overflowing vessel, and the weight of the overflow is equal to the buoyant force.

We achieved this measurement by placing a large beaker on an electronic scale. We then zeroed the scale with the beaker on it. We then placed a cup nearly filled with water into the beaker and submerged the cylinder.

The first attempt did not result in any overflow, due to the surface tension of the water.

On the second attempt, we filled the cup until the surface tension brought the water line above the edge. Once the cylinder was submerged, an amount of water flowed into the larger beaker. we then removed the cup and cylinder from the beaker and used the scale to measure the mass of the water in the beaker.

The mass displaced was recorded as 6.8 grams and this translates to a weight of .0666 ±.005 Newtons.

This differs from our first measurement by approximately 4± 4%

The final method for determining the buoyant force was to measure the volume of the cylinder using vernier callipers.

Our cylinder's diameter was recorded to be .0143 ± .0001 meters, the height was recorded to be .0484 ± .0001 meters. By the formula for the volume of a cylinder:

Our Volume was determined to be approximately:
7.77±.01 *10^-6 meters cubed.

the density of Deionised water at 23 degrees Celsius, (the temperature of the laboratory at the time of the experiment) is approximately 997.568 kg/m^3.
This results in a mass of displaced water approximately:
.00775442 kg

The weight of that water would be .0761±.0001 Newtons.

That is a 16±1% increase over the reading from the force sensor and a 14±1% increase over the prediction from the scale test.

Despite this calculation's significant deviation from the previous two values, it is in all likelihood the most accurate. The scale and force sensor may not have been properly calibrated, and the experimental procedure was rushed. On the scale trial, there may have been some water still adhering to the cup as it was withdrawn from the beaker. The force sensor was only casually calibrated, and may not have high enough resolution for the scale of the measurements we were taking.

The vernier callipers are at least high enough quality that we can be confident about the measurements up to the hundredth of a centimeter, this represents less than one percent uncertainty in that measurement.


-B