Experiment 10: Measuring a Human Hair

Introduction:

The purpose of this experiment is to measure the thickness of a human hair by applying Young's double slits experiment.

Data:

A strand of hair is used as a slit in the experiment

Setup of apparatus used for experiment

Using a red laser (660nm wavelength), an interference pattern was projected onto a white board.

Interference Pattern

Calculations of thickness of the hair

where: y = distance between 2 fringes

            L = distance between the card and the white board

            d = width of the slit

We then used a micrometer to measure the thickness of the hair and verify the calculated value.

Measuring the thickness using a micrometer.

Micrometer measurement: 50 ± 20μm

Conclusion:
The Young's double slit experiment can be used to measure the thickness of human hair. The percent error was 65%, which falls within the range of uncertainty.

Experiment 14: Classical Harmonic Oscillator

Potential Energy Diagrams: A particle of energy 12 x 10^-7 J moves in a region of space in which the potential energy is 10 x 10^-7 J between the points -5 cm and 0 cm, zero between the points 0 cm and +5 cm, and 20 x 10^-7 J everywhere else.

Question 1: What will be the range of motion of the particle when subject to this potential energy function?

The range of motion is between +-5 cm.

Question 2: Clearly state why the particle can not travel more than 5 cm from the origin.

The energy that the particle has is less than the energy at the top of the well.

Question 3: Assume we measure the position of the particle at several random times. Is there a higher probability of detecting the particle between -5 cm and 0 cm or between 0 cm and +5 cm.

There is a higher probability of finding the particle from -5 to 0 cm because it has a higher potential energy on that side.

Question 4: What will happen to the range of motion of the particle if its energy is doubled?

The range of motion of the particle will increase.

Question 5: Clearly describe the shape of the graph of the particle's kinetic energy vs. position.

The graph is an inverted parabola 

Question 6: Assume we measure the position of the particle at several random times. Where will the particle most likely be detected?

The particles will most likely be found at the ends where the boundaries are located.

Potential Wells: A particle is trapped in a one-dimensional region of space by a potential energy function which is zero between positions zero and L, and equal to U₀ at all other positions. This is referred to as a potential well of depth U₀. Examine a proton in a potential well of depth 50 MeV and width 10 x 10^-15 m.

Question 1: If the potential well was infinitely deep, determine the ground state energy. Is this also the ground state energy in the finite well?

Energy in infinite well: 2.05 MeV

Energy in finite well: 1.8MeV

The energy in the infinite well is greater than the energy in the finite well. 

Question 2: If the potential well was infinitely deep, determine the energy of the first excited state (n = 2). Is this also the energy of the first excited state in the finite well?

Energy in first excited state of infinite well = 8.4 MeV

Energy in first excited state of finite well = 6.8 MeV

The energy of the first excited state in the infinite well is greater than the energy of the first excited state in the finite well.

Question 3: Since the wavefunction can penetrate into the "forbidden" regions, will the energy of the first excited state in the finite well to be greater than or less than the energy of the first excited state in the infinite well? Why? 

The energy in the first excited state of the finite well is less than the energy of the first excited state in the infinite well. The is because in the finite well, there is a greater probability of tunneling than there is in the infinite well.

Question 4: Will the energy of the n = 3 state increase or decrease if the depth of the potential well is decreased from 50 MeV to 25 MeV? Why? 

The energy of the n=3 state decreases if the potential well is decreased from 50MeV to 25MeV. This is because there is less of a probability tunneling.

Question 5: What will happen to the penetration depth as the mass of the particle is increased?

As the mass of the particle is increased, the penetration depth decreases.

Experiment 9: Lenses

Introduction: The purpose of this experiment is to investigate the relationship between object distance, image distance, and the focal length of a lens.

Data:
The focal length of a lens was measured by taking the lens outside, using the sunlight to create an image on an object, and measuring the distance between the lens and the focused image.

Set up of the lab at the initial position, the measured focal point multiplied by 5.

 Our experiment has us move the lens in to points of 5f, 4f, 3f, 2f, and 1.5f.

A focused image.

The image at half the focal length. The image becomes virtual.

If someone were to look through the lens at the object, which is the light, they would see an image that is much larger and oriented upright.

Data from experiment

Graph for data of object distance vs. image distance.

Table of data for inverse object distance and inverse image distance.

Graph of inverse values.

Looking at the graph of the inverse values, we can see that we are given a linear equation (y = mx + b), which we can model closely to the equation:

From here, we observe that our y-intercept on the graph is related to our inverse focal length. 

Focal length from graph = 1/.1098 = 9.11 cm.

Measured focal length = 8 cm. 

% error = (9.11-8)/9.11 * 100 = 12.2%

Our error could come from the fact that it was a cloudy day, so measuring focal length using sunlight was very difficult. Measurements taken could have been off.

Experiment 8: Concave and Convex Mirrors

Introduction: In this lab we observe the visual effects given by varying distances when looking into a concave and convex mirror.

Data:
Convex mirror

Close up view of convex mirror

Far away view of convex mirror.

Characteristics of image formed on a convex mirror:
1a. Image appears smaller than the object
  b. Image is upright
  c. Image distance seems to be larger than the object distance

2. When moved closer, the image still exhibits the same characteristics.

3. As the object get further away from the mirror, the image size becomes a lot smaller.

Distance Object = 5.5 centimeters

Distance Image = 1.9 centimeters

Height Object = 2.2 centimeters

Height Image = 0.7 centimeters

Magnification = height image/height object = 0.32

Concave mirror

Close up view of concave mirror.

Far away view of concave mirror.

Characteristics of image formed by concave mirror:
1a. The image appears larger than the object
  b. The image is upright
  c. The image distance is smaller than the object distance

2. As the object moves farther, the image appears smaller than the object and it becomes inverted.

Distance Object = 11.6 centimeters

Distance Image = 3.3 centimeters

Height Object = 2.3 centimeters

Height Image = 0.6 centimeters

Magnification = height image/height object = 0.26

Results/Conclusion:

The image formed by a convex mirror is upright but smaller, as opposed to the concave mirror which forms an image that is upside down and smaller when behind the focal point, while being in front of the focal point forms and image that is upright and bigger than the original object.

Experiment 6: Electromagnetic Radiation

Introduction:
This lab was conducted in order to observe electromagnetic radiation through the use of an antenna and a oscilloscope.

Data:

Set up of the experiment.

An example of the oscilloscope when the antenna is at 0.1 meters from the receiver.

An example of the oscilloscope when the antenna is 0.3 meters from the receiver.

Table of data recorded from measurements on oscilloscope.

Our graph with an A/r fit.

Our graph with an A/r^2 fit.

As we can see from the graphs, our data best fits an A/r graph.

Results/Conclusion:

Theoretical Analysis:

L = 0.1 Meters

Q = 1.24 * 10 ^ -10 Coulombs

V_0 = 40.29 mV

From calc varient: 

  V=  2kQ/L (ln|L/2+√(L^2/4+z^2 )|-ln|z|) + Vo

V =  2k(1.24*10 ^ -10)/0.1 (ln|0.1/2+√(0.1^2/4+z^2 )|-ln|z|) + 40.29

Theoretical voltage as distance increases.

Graph of theoretical voltage vs. distance.

Comparing the theoretical values to the experimental values:

Comparison graph between the two sets of voltage values vs distance.

The value sets are around the same, but there are obvious errors in the measurements taken.

Uncertainty Analysis:

Voltage uncertainty: +- 10 mV for 50 mV

                              +- 5 mV for 20 mV

                              +- 2 mV for 10 mV

Distance uncertainty: Operational error = +- 0.0001 meters

                               Systematic error = +- 0.02 meters

Total uncertainty: 0.021 meters +- 0.001 meters

Theoretical voltages using uncertainty from distance measurements.

Graph of theoretical voltage maximums and minimums.

Graph of peak to peak voltage with uncertainties.

Comparison of theoretical maximums and minimums with peak to peak voltages with uncertainties.

Comparing the theoretical and experimental graphs, we can see that our graphs don't agree with each other. 

Discussion:
Through this experiment, we found that frequency, distance, and voltage all affect the  magnitude of the electromagnetic wave radiation. Our uncertainty values could be given by bad measurements and faulty equipment. We found that other devices placed near the oscilloscope affected the reading on the oscilloscope, such as cellular phones or other devices that give off electromagnetic radiation.

Experiment 7: Introduction to Reflection and Refraction

Introduction: The purpose of this lab is to observe the laws of refraction and reflection through two different mediums with different indices of refraction.

Data:
First Experiment:

Set up of the lab for the first experiment

First trial using flat side of semi-circular prism

Example of reflection and refraction at some angle using flat surface of semi-circular prism

Refraction angles collected at various degrees

Plot of angle 1 vs. angle 2

Plot of the sine values of each angle

 Second experiment:

Second experiment conducted where the curved side of the semi-circular prism was used.

Example of total internal reflection at a certain angle that was calculated to be about 42 degrees.

Refraction angles measured at various degrees. After a certain point, the light begins to completely reflect inside the semi-circular prism

Plot of angle 1 vs angle 2 for the second experiment where light entered through the curved side of the semi-circular prism

Plot of the sine values of each angle for the second experiment

Results/Conclusion:

In our first experiment, we found that there is a linear relationship between the angle of incidence (Theta 1) and the angle of refraction (Theta 2). In the second experiment, we found the same relationship between angle of incidence and angle of refraction, but after we reached a certain angle of about 40 degrees, we found that our light experienced total internal reflection. This phenomena happens after the angle of refraction reaches a point where it is reflected along the side of the prism. The light experiences no refraction and instead complete reflection. Looking at the angle of our second graph for trial #2, we also found that the slope of the graph gives us the index of refraction of our semi-circular prism which is made out of acrylic. Our graph gives us an index of 1.4875, while the accepted value is 1.49.