OVERVIEW
This lab series is very similar to the Physics 6A and 6B series. Please refer to those lab manuals for general information.
Important: This manual assumes that you have already taken the Physics 6A and 6B labs, and that you are familiar with Microsoft Excel. In addition, it assumes that you are able to perform all the operations associated with Data Studio (particularly for Experiment 5); call up sensors; use them to take various measurements; produce, title, label, and vary the appearance of graphs; perform calculations on the measured variables in Data Studio, and use the results to create graphs.
Note to TAs: You should have taught a Physics 6A lab section before teaching a 6C lab. If you have not, you should make sure that you have gone through all the Data Studio operations for an experiment (particularly Experiment 5) before teaching it.
Note to Instructors: The thermodynamics experiment, Experiment 5, requires two lab sessions to complete. It consists of two parts: a measurement of absolute zero using the Ideal Gas Law, and an experiment with a heat engine. For the experiment to be assigned two sessions, you would have to make the request at the beginning of the quarter, and possibly omit the radioactivity or photoelectric experiment. If you take no action, the default option is that the experiment will be assigned one session, and the students will just do the absolute zero measurement. (Later in the quarter up to the week before the experiment, you could request that the students do the heat engine part of the experiment instead of the absolute zero measurement.
It is essential that you follow the general rules about taking care of equipment and reading the lab manual before coming to class.
As before:
Lab grade = | (12.0 points) |
− (2.0 points each for any missing labs) | |
+ (up to 2.0 points earned in mills of “additional credit”) | |
+ (up to 1.0 point earned in “TA mills”) |
Maximum score = | 15.0 points |
Typically, most students receive a lab grade between 13.5 and 14.5 points, with the few poorest students (who attend every lab) getting grades in the 12s and the few best students getting grades in the high 14s or 15.0. There may be a couple of students who miss one or two labs without excuse and receive grades lower than 12.0.
How the lab score is used in determining a student's final course grade is at the discretion of the individual instructor. However, very roughly, for many instructors a lab score of 12.0 represents approximately B− work, and a score of 15.0 is A+ work, with 14.0 around the B+/A− borderline.
POLICY ON MISSING EXPERIMENTS
In the Physics 6 series, each experiment is worth two points (out of 15 maximum points). If you miss an experiment without excuse, you will lose these two points.
The equipment for each experiment is set up only during the assigned week; you cannot complete an experiment later in the quarter. You may make up no more than one experiment per quarter by attending another section during the same week and receiving permission from the TA of the substitute section. If the TA agrees to let you complete the experiment in that section, have him or her sign off your lab work at the end of the section and record your score. Show this signature/note to your own TA.
(At your option) If you miss a lab but subsequently obtain the data from a partner who performed the experiment, and if you complete your own analysis with that data, then you will receive one of the two points. This option may be used only once per quarter.
A written, verifiable medical, athletic, or religious excuse may be used for only one experiment per quarter. Your other lab scores will be averaged without penalty, but you will lose any mills that might have been earned for the missed lab.
If you miss three or more lab sessions during the quarter for any reason, your course grade will be Incomplete, and you will need to make up these experiments in another quarter. (Note that certain experiments occupy two sessions. If you miss any three sessions, you get an Incomplete.)
APPARATUS
Shown in the pictures below:
Not shown in the pictures above:
NOTE TO INSTRUCTORS
This experiment consists of two parts. Part 1 involves checking the magnetic field produced by a current loop, while part 2 is an investigation of Faraday's Law. Most students cannot complete these two parts in one lab session, so you should choose which part you souls like them to perform. The default option (in which you do not express a preference) is part 2.
MAGNETIC FIELDS
The most basic principle of electricity and magnetism is that charges exert forces on other charges. This picture is very simple if the charges are stationary: only the Coulomb force is present. Rather than describing the forces as action at a distance, however, we will use the field picture, whereby one charge creates a field, and other charges in the field feel forces from the field. If the charges are stationary, then only electric fields are involved; but if the charges are moving, magnetic fields also come into play. The field picture is used because the fundamental equations of electricity and magnetism — Maxwell's Equations — are much simpler when written in terms of fields than in terms of forces.
If a charge \(q\) is stationary, then it creates only an electric field \(\textbf{E}\):
\begin{eqnarray} \textbf{E} &=& q\textbf{r}/4\pi\varepsilon_0 r^3. \label{eqn_1} \end{eqnarray}
A charge moving at velocity \(v\) also creates a magnetic field \(\textbf{B}\):
\begin{eqnarray} \textbf{B} &=& \mu_0 q\textbf{v}\times\textbf{r}/4\pi r^3. \label{eqn_2} \end{eqnarray}
If a test charge is stationary, then it feels only an electric force \(\textbf{F}_E\):
\begin{eqnarray} \textbf{F}_E &=& q\textbf{E}. \label{eqn_3} \end{eqnarray}
If the test charge is moving, then it also feels a magnetic force \(\textbf{F}_B\):
\begin{eqnarray} \textbf{F}_B &=& q\textbf{v}\times\textbf{B}. \label{eqn_4} \end{eqnarray}
Often for the case of magnetic fields, the moving charges are part of an electric current. Consider a current \(i\) of charges \(q\) moving at an average drift velocity \(v_0\).
If the number of charges per unit volume in the material is \(n\), then the total charge \(\Delta q\) in the cylindrical volume of cross-sectional area \(A\) and length \(\Delta L = v_0\Delta t\) is \(nqA\Delta L\), since the volume of the element is \(A\Delta L\). Thus, the current \(i\) which passes through one cap of the cylindrical element is
\begin{eqnarray} i &=& \Delta q/\Delta t = nqA \Delta L / \Delta t = nqA(v_0\Delta t)/\Delta t = nqAv_0. \label{eqn_5} \end{eqnarray}
To find the magnetic field produced by this element of wire, we need to sum the \(qv_0\) terms in Eq. \eqref{eqn_2}:
\begin{eqnarray} \textbf{B} &=& \mu_0 \left(\sum q\textbf{v}_0\right) \times \textbf{r}/4\pi r^3. \end{eqnarray}
In the cylindrical element, the total number of charges is \(nA\Delta L\), so
\begin{eqnarray} \sum q\textbf{v}_0 &=& (nqA \Delta L) \textbf{v}_0 = i \Delta\textbf{L}, \end{eqnarray}
where we have used Eq. \eqref{eqn_5} in the last step, and have noted that \(\Delta\textbf{L}\) is in the same direction as \(\textbf{v}_0\). Thus, the magnetic field produced by this element of wire is
\begin{eqnarray} \Delta\textbf{B} &=& \mu_0 (i\Delta \textbf{L}) \times \textbf{r}/4\pi r^3. \label{eqn_6} \end{eqnarray}
Eq. \eqref{eqn_6} is the Law of Biot and Savart. (We call the magnetic field from this element \(\Delta\textbf{B}\) since there must be additional contributions to \(\textbf{B}\) from other parts of the wire carrying the current.)
MAGNETIC FIELD AT THE CENTER OF A LOOP OF WIRE
Consider a circular wire loop of radius \(R\) and carrying a current \(i\). We are interested in the magnetic field at the center of the wire. (Of course, a real loop would need to be interrupted by a battery at some point to keep the current flowing through the resistance of the wire, unless the wire were a superconductor.)
For the current loop, the cross product \(\Delta\textbf{L} \times \textbf{r}\) is perpendicular to the plane of the loop, so all the \(\Delta\textbf{L}\) terms contribute to the magnetic field in the same direction. Note that in this case, the magnitude of \(\textbf{r}\) (the vector from the current element to the observation point) is equal to \(R\) (the radius of the loop). Therefore, to obtain the total magnetic field at the center of the loop, we need only to add all the \(\Delta\textbf{L}\) terms; no other quantities change as we move around the circle:
\begin{eqnarray} \int\textrm{d}L &=& 2\pi R. \end{eqnarray}
The magnitude of \(\textbf{B}\) for a loop is thus
\begin{eqnarray} B &=& \int \textrm{d}B = \mu_0 i \left(\int\textrm{d}L\right) R/4\pi R^3 = \mu_0 i (2\pi R) R/4\pi R^3 = (\mu_0/4\pi) (2\pi i/R), \label{eqn_7} \end{eqnarray}
and the direction of \(\textbf{B}\) is perpendicular to the plane of the loop. If you curl the fingers of your right hand in the direction that the current (which, by convention, is positive) flows, then your thumb will point in the direction of \(\textbf{B}\).
For a loop of \(N\) turns, the current is \(N\) times the current in one turn, so the magnitude of \(\textbf{B}\) would be
\begin{eqnarray} B &=& (\mu_0/4\pi) (2\pi Ni/R). \label{eqn_8} \end{eqnarray}
Now let us find the value of \(\textbf{B}\) at a distance \(z\) along the axis perpendicular to the plane of the loop.
Here the position vector \(\textbf{r}\) from the current element \(\textrm{d}\textbf{L}\) to the observation point a distance \(z\) along the axis is the diagonal vector whose magnitude is \((z^2+R^2)^{1/2}\). The magnetic field contribution \(\textrm{d}\textbf{B}\) is at an angle \(\theta\) with respect to the \(z\)-axis. As we integrate around the circle, only the vertical (\(\cos\theta\)) components add up; the horizontal (\(\sin\theta\)) components are canceled by an equal contribution on the opposite side. Also note that the angle \(\alpha\) is equal to \(\theta\), from the geometry theorem that if two lines meet at an angle, then two other lines, each perpendicular to one of the first two lines, make the same angle. Thus, \(\cos\theta = R/(z^2+R^2)^{1/2}\), and the contribution of the magnetic field along the \(z\)-axis is
\begin{eqnarray} \textrm{d}B_z &=& \textrm{d}B\,\cos\theta = (\mu_0/4\pi) (Ni\textrm{d}L/r^2) \cos\theta = (\mu_0/4\pi) [Ni\textrm{d}L/(z^2+R^2)] [R/(z^2+R^2)^{1/2}]. \end{eqnarray}
Again, when we integrate around the loop, none of the other quantities change, so using \(\int\textrm{d}L = 2\pi R\), we obtain
\begin{eqnarray} B &=& (\mu_0/4\pi) [2\pi NiR^2/(z^2+R^2)^{3/2}]. \label{eqn_9} \end{eqnarray}
You should be able to verify quickly that Eq. \eqref{eqn_9} reduces to Eq. \eqref{eqn_8} for \(B\) at \(z = 0\).
FARADAY'S LAW
Conceptually, Faraday's Law tells us that changing magnetic fields induce electric fields. Mathematically, this law states that the emf \(\mathcal{E}\) — the integral of the electric field around a closed path — is equal to the change in magnetic flux \(\Phi\) through the path:
\begin{eqnarray} \mathcal{E} &=& \int \textbf{E}\cdot\textrm{d}\textbf{l} = -\textrm{d}\Phi/\textrm{d}t, \label{eqn_10} \end{eqnarray}
where
\begin{eqnarray} \Phi &=& \int \textbf{B}\cdot\textrm{d}\textbf{A} \end{eqnarray}
The minus sign in Eq. \eqref{eqn_10} reminds us of Lenz's Law: the emf is induced in such a direction as to oppose the change in magnetic flux that produced it.
In this experiment, we will be testing Faraday's Law by monitoring the emf induced in a small search coil of \(N\) turns, positioned in a changing magnetic field. For such a coil, the emf will be \(N\) times larger than the emf induced in one turn:
\begin{eqnarray} \mathcal{E} &=& -N\,\textrm{d}\Phi/\textrm{d}t. \end{eqnarray}
Furthermore, if the search coil is small enough so that \(\textbf{B}\) can be considered constant over the area, then
\begin{eqnarray} \Phi &=& \int \textbf{B}\cdot\textrm{d}\textbf{A} = \textbf{B}\cdot \int\textrm{d}\textbf{A} = \textbf{B}\cdot\textbf{A}. \end{eqnarray}
Combining these results, we obtain the version of Faraday's Law which will be tested in this experiment:
\begin{eqnarray} \mathcal{E} &=& -(\textrm{d}/\textrm{d}t) (\textbf{B} \cdot N\textbf{A}). \label{eqn_11} \end{eqnarray}
The emf \(\mathcal{E} = \int \textbf{E}\cdot\textrm{d}\textbf{l}\) is quite similar to the potential difference \(\Delta V = \int \textbf{E}\cdot\textrm{d}\textbf{l}\), and can be measured with a voltmeter or the voltage sensor of Data Studio.
PROCEDURE — CHECKING THE MAGNETIC FIELD OF A CURRENT LOOP
We will be using the Rotary Motion Sensor with the Linear Motion Accessory to map the axial field of a current coil. Arrange your apparatus as shown in the diagram below. We are not using the second coil until step 8. It may be present, but should not be hooked to a power supply yet.
Double-click on the Rotary Motion Sensor in Data Studio, and insert its plugs into the appropriate digital channels. Also call up the Magnetic Field Sensor in the setup window, and insert the physical plug into analog channel A.
For magnetic field measurements, the coil draws power from the DC power supply, which is wired through the Fluke Multimeter to measure the current to the coil. Be sure to turn the coarse and fine voltage controls of the power supply to zero before switching on the power supply; otherwise, the initial current may be too large and blow the fuse in the Multimeter. To measure a current, the Multimeter must be in series with the power source. Wire one lead from the ground of the DC power supply to the “Common” plug of the Multimeter. Wire a second lead from the mA plug of the Multimeter to one of the coil plugs. The second coil plug is wired back to the other output plug of the power supply. Wire directly to the coil leads; do not use the built-in series 1.2-kΩ resistor in this part of the experiment. Set your Multimeter to read on the 2000-mA DC scale.
Set the voltage control of the power supply to zero, turn on the power supply, and adjust the coarse and fine voltage controls slowly until you obtain a current of 1000 mA = 1A.
Make a preliminary check of the magnetic field of the coil. Carefully position the end of the Magnetic Field Sensor at the center of the coil. (We are not making computer use of the rotational sensor yet.) Use the controls on the Magnetic Field Sensor (hardware) to set it to radial mode and 10× reading. Double-click on the Magnetic Field Sensor on the computer screen, and check that the 10× measurement appears in the “Data” column. Drag a digits window to the Magnetic Field Sensor, and drag the 10× data to it. Turn “OFF” the current to the coil from the power supply, and click “Start” to see the gauss reading of the sensor. Push the “Tare” button on the sensor to zero the magnetic field reading. It is a good idea always to zero the Magnetic Field Sensor with zero current before recording measurements with the current on. Even when zeroed, the reading may jump around a bit. Now turn the power supply back on, and record the magnetic field reading and the current. (If your magnetic field measurements have negative values and you don't like this, then reverse the leads to the power supply.)
Magnetic field (gauss) =
Current (mA) =
Compare your measured magnetic field with the calculated field from \(B = (\mu_0/4\pi) (2\pi Ni/R)\) (Eq. \eqref{eqn_8}). Remember that \(\mu_0/4\pi\) = 10-7 Tm/A, and you can read off the value of \(N\) (the number of turns) and \(R\) (the radius of the coil). Be sure to put \(R\) and \(i\) in the proper SI units. This formula gives \(B\) in teslas; convert to gauss using 1 tesla = 104 gauss.
Calculated magnetic field (gauss) =
Percentage error =
Now arrange the stand with the Magnetic Field Sensor to map the magnetic field of the coil along its axis. Your arrangement should be such that as you turn the Rotary Motion Sensor, the Magnetic Field Sensor starts on one side of the coil, passes through the center of the coil, and moves beyond the other side — always staying on the axis of the coil. Double-click on the Rotary Sensor icon, and check the position measurement so it shows up in the “Data” column. On the computer screen, drag a graph to the Rotary Motion Sensor. Drag the position data to the \(x\)-axis of the graph, and the magnetic field data to the \(y\)-axis. Make a trial run by turning off the current, zeroing the magnetic field reading with the “Tare” button, turning on the current, clicking “Start”, and moving the sensor through the field by rotating the pulley wheel on the Rotary Motion Sensor. Rotate smoothly — although the speed is not important, since we are not plotting anything as a function of time. You should see a nice graph of the magnetic field plotted against the axial distance.
After any readjustments, when you have a nice plot on the computer, you may print it out. DO NOT DISCARD THE DATA. Save it in a file, which you can retrieve, on the desktop. You may use it later for additional credit.
Now wire the second coil in parallel with the first. You want the currents in the coils to flow in the same direction, so the magnetic fields of the two coils add in the space between the coils. Again, adjust the voltage of the power supply so the current to each coil is 1 A (with a total current of 2 A). Arrange the stand with the Magnetic Field Sensor to move along the axis of both coils, particularly covering the area between them. Take three measurements of \(B\) versus axial distance with the coil separations equal to \(0.5 R\), \(1.0 R\), and \(1.5 R\), where \(R\) is the radius of the coils. (In order to make the three graphs comparable, start each measurement with the end of the Magnetic Field Sensor at the center of one coil. If the leads of the second coil prevent you from moving it to the \(0.5 R\) distance from the first coil, how can you manipulate the apparatus to make this measurement possible?) Arrange a graph on the computer so that all three plots appear on the same graph aligned vertically. You may print this page out for your records.
What coil separation produces the most uniform magnetic field between the coils?
This arrangement is called Helmholtz coils, and is a method of producing a relatively constant, controllable magnetic field over a considerable volume of space. Of course, we have just measured the magnetic field along the axis, but the field is fairly uniform throughout much of the volume of space between the coils. We used this property of Helmholtz coils in the previous \(e/m\) experiment.
PROCEDURE — FARADAY'S LAW
Disconnect and set aside the power supply, the multimeter, and the stand with the Magnetic Field Sensor. Start with a new Data Studio display, being sure to save the data you might need for the additional credit.
Wire the signal generator output of the Science Workshop interface to one of the coils, including the 1.2-kΩ resistor in series. We will call this the “field coil”. Since the resistance of the coil itself is negligible compared to the 1.2 kΩ of the resistor, the voltage of the signal generator divided by 1.2 kΩ gives the current through the field coil.
Wire a voltage sensor (just an analog plug with two leads) to the 2000-turn search coil, and set it up in Data Studio. The search coils have 10-kΩ resistors in parallel to damp out oscillations. On the computer screen, double-click “Signal Output” to get the output voltage into the data column and bring up the signal generator control window. Set up a scope display with two traces to measure both the voltage output of the signal generator and the output of the voltage sensor. Select the signal generator voltage so the scope triggers on this signal.
Set the signal generator to a 5-V triangle wave at 2000 Hz, turn it on, click “Start”, and make any adjustments to the scope so that you see a clear, stable triangle wave on the scope. Check the trigger setting if the wave on the scope is not stable. Adjust the trace speed by clicking the arrows on the \(x\)-axis of the scope, and the amplitude of the trace by clicking the appropriate arrows on the \(y\)-axis. Now when you bring up the search coil inside the field coil and in the same plane, and adjust the voltage sensor signal on the scope to greater sensitivity, you should see a square wave trace, possibly with some fluctuations. Notice that when you click “Stop”, the last scope trace remains fixed in the scope window, facilitating some of the measurements below.
Faraday's Law (Eq. \eqref{eqn_11}) states that \(\mathcal{E} = -(\textrm{d}/\textrm{d}t) (\textbf{B} \cdot N\textbf{A})\), where \(\mathcal{E}\) is the emf induced in the search coil (which you are measuring with the voltage sensor). \(\textbf{B}\) is the magnetic field of the field coil, which is proportional to the current flowing through it; this current, in turn, is proportional to the triangle wave voltage from the signal generator. \(\textbf{A}\) is the area vector of the search coil, and \(N\) is the number of turns of the search coil. The cosine of the dot product between \(\textbf{B}\) and \(\textbf{A}\) is 1.0 if the search coil is in the same plane as the field coil.
Study your two scope traces, and explain clearly, briefly, and neatly the shape and the phase of the voltage sensor output using the equation above.
PROCEDURE — DEPENDENCE OF \(\mathcal{E}\) ON \(\textrm{d}i/\textrm{d}t\)
As you can see from Faraday's Law, \(\mathcal{E}\) is proportional to \(\textrm{d}i/\textrm{d}t\) (where \(i\) is the current through the field coil), since \(B\) of the field coil is proportional to \(i\). In turn, \(\textrm{d}i/\textrm{d}t\) is related to the frequency for a triangular wave. Write down the correct equation relating \(\textrm{d}i/\textrm{d}t\) to the frequency \(f\) and amplitude \(i_{\textrm{m}}\) for a triangular wave.
Check this relation. Measure \(\mathcal{E}\) (by putting the smart tool on the square wave after you click “Stop”) at 500, 1000, 1500, 2000, and 2500 Hz. Enter your data in a new Excel worksheet, chart \(\mathcal{E}\) as a function of \(f\) with a clear title and labels on the axes. You may print out the graph for your records. Be sure you position the search coil in the same place at the center of the field coil when making each measurement.
DEPENDENCE OF \(\mathcal{E}\) ON THE DOT PRODUCT WITH THE AREA VECTOR
Devise a way to use the EZ angle protractor to adjust the dot product angle in \(\textbf{B} \cdot N\textbf{A}\) between the area vector of the search coil and the magnetic field (perpendicular to the field coil plane). The search coil should be at the center of the field coil for each measurement. It may help to have both lab partners holding parts of the equipment. Remember that when you click “Stop”, the signal will remain on the scope for easy measurement with the smart tool.
Keeping the frequency and amplitude of the triangle wave constant, measure \(\mathcal{E}\) for dot product angles of 0°, 30°, 45°, 60°, and 90°. Record your data in a new Excel worksheet, compute the cosines with a fill-down operation, and chart \(\mathcal{E}\) as a function of the cosine of the angle. Title the chart and label the axes clearly. You may print it out for your records.
PROCEDURE — DEPENDENCE ON \(N\)
We have been using the 2000-turn search coil up to now. Measure \(\mathcal{E}\) with a triangle wave of amplitude 5 V and frequency 2000 Hz, using the 2000-turn search coil at the center of the field coil and in the same plane. Now, what should \(\mathcal{E}\) be with a 400-turn search coil? Repeat the measurement with the 400-turn coil, and record your result below.
Predicted \(\mathcal{E}\) with 400 turns =
Measured \(\mathcal{E}\) with 400 turns =
PROCEDURE — SINE WAVES
If the current to the field coil is a sine wave, what form of wave (including phase information) should the induced \(\mathcal{E}\) take? Verify this with your equipment.
ADDITIONAL CREDIT (5 mills maximum)
Before you start the additional credit, check that you have the following graphs with good data, titles, and labeled axes:
Magnetic field versus axial distance for a single coil.
Three magnetic field measurements versus axial distance for three different coil separations plotted on the same graph.
An Excel chart of \(\mathcal{E}\) versus \(f\).
An Excel chart of \(\mathcal{E}\) versus the cosine of the angle of the dot product.
The magnetic field on the axis at a distance \(z\) from a current coil of radius \(R\), \(N\) turns, and current \(i\) is given by Eq. \eqref{eqn_9}:
\begin{eqnarray} B &=& (\mu_0/4\pi) \left[ 2\pi NiR^2/(z^2+R^2)^{3/2} \right]. \end{eqnarray}
Go back to the data that you saved earlier for the magnetic field along the axis of the coil. Set up a calculation in Data Studio for the equation above, and plot it on the same graph as the magnetic field data so you can compare the theoretical magnetic field with the measured field. The reward is 5 mills if you have a neatly labeled printout, done without significant assistance from your TA. If you need assistance, you can “buy” varying amounts by giving up some of the 5 mills. Here are a couple of hints: \(B\) in the equation above is in teslas, but the measured field is in gauss, so include the conversion to gauss in your calculation of the theoretical field. You will also need to change the origin of the \(z\)-axis in the calculation to match the position of the center of the coil for your measured field.
APPARATUS
INTRODUCTION
In this experiment, you will test several optical aspects of electromagnetic waves such as polarization, reflection, and interference. The electromagnetic spectrum covers a wide range of frequencies. Visible light has a frequency of the order of 1014 Hz and wavelengths between 400 and 700 nm (1 nm = 10−9 m). Other well-known parts of the spectrum include radio waves (with frequencies near 106 Hz) and microwaves (with frequencies around 1010 Hz and wavelengths of a few centimeters). Microwaves can be generated easily and are particularly suited for laboratory investigations.
POLARIZATION, REFLECTION, AND ABSORPTION OF MICROWAVES
Electromagnetic waves consist of position-dependent and time-dependent electric and magnetic fields which are perpendicular to each other. These waves propagate in a direction perpendicular to both fields. In this experiment, we consider microwaves produced by a transmitter whose axis is vertical. The electric fields of these microwaves are therefore linearly polarized in the vertical plane and travel in the horizontal direction.
A receiver which detects such microwaves measures only the component of the incident electric field parallel to its axis. If the angle between the incident electric field (of amplitude \(E_0\) and the receiver axis is \(\theta\), then the parallel component of the field has amplitude \(E_0\cos\theta\), as shown in the figure above. Since the intensity of a wave is proportional to the square of its amplitude, the intensity \(I\) measured by the receiver is related to the intensity \(I_0\) of the incident wave by
\begin{eqnarray} I &=& I_0 \cos^2\theta. \label{eqn_Malus} \end{eqnarray}
Eq. \eqref{eqn_Malus} is known as Malus' Law and tells us how the intensity varies with angle between the transmitter and receiver.
A wave incident on a metallic surface will be reflected after striking the surface. The law of reflection states that the angle of reflection \(\theta_{\textrm{r}}\) is equal to the angle of incidence \(\theta_{\textrm{i}}\): \(\theta_{\textrm{r}} = \theta_{\textrm{i}}\). Note that both angles are measured with respect to the normal to the surface.
Microwaves which impinge upon an opaque material are either reflected by, transmitted through, or absorbed by the material. Let us denote the reflected, transmitted, absorbed, and total electric-field amplitudes by \(R\), \(T\), \(A\), and \(E\), respectively. The law of energy conservation tells us that the total energy of the incident microwaves is equal to the sum of the reflected, transmitted, and absorbed energies. Since energy is directly proportional to intensity and therefore proportional to the square of the electric-field amplitude, it follows that
\begin{eqnarray} E^2 &=& R^2 + T^2 + A^2 \end{eqnarray}
or
\begin{eqnarray} A^2 &=& E^2 - R^2 - T^2. \label{eqn_Absorbed} \end{eqnarray}
Thus, the percentage \(f_{\textrm{A}}\) of the incident microwave intensity absorbed by a material is
\begin{eqnarray} f_{\textrm{A}} &=& [(E^2 - R^2 - T^2) /E^2] \times 100\%. \label{eqn_fAbsorbed} \end{eqnarray}
INTERFERENCE
When two separate waves occupy the same region of space, they combine with each other. According to the superposition principle, the displacement of the resultant wave is equal to the sum of the displacements of the individual waves. If the crests of the individual waves coincide with each other, then the amplitude of the resultant wave is a maximum, and the waves are said to undergo constructive interference. On the other hand, if the crest of one wave coincides with the trough of the other wave, then the amplitude of the resultant wave is zero at all points, and the waves undergo destructive interference. Waves that interfere constructively “build each other up” and have a maximum intensity, while those that interfere destructively “cancel each other out” and have a minimum intensity.
In this experiment you will build a device called a Michelson interferometer that splits a wave into two waves and then recombines the waves after they have traveled different distances. If the extra distance traveled by one of the two waves (called the path difference) is equal to an integral multiple of one wavelength (i.e., 0, \(\lambda\), \(2\lambda\), etc.), constructive interference results, and the combined waves be measured to have a large intensity, as shown in the figure above. Conversely, if the path difference is equal to an odd integral multiple of a half wavelength (i.e., \(\lambda/2\), \(3\lambda/2\), \(5\lambda/2\), etc.), destructive interference occurs, and the waves will cancel when they overlap and produce zero intensity.
INITIAL SETUP
You may find one of two types of microwave receiver/transitter setups at your station. Both of these setups use Gunn diodes to generate microwaves.
The most modern version, Pasco WA 9314B, is the easiest to use, and has a self-contained meter. Align the receiver horn facing the transmitter horn, and plug the transmitter in to turn it on. On the receiver, turn the intensity knob from “off” to “30”. Adjust the variable sensitivity knob for a full scale reading. You can increase the sensitivity later in the experimet if needed. When finished with the experiment, unplug the transmitter, and turn off the receiver.
The older Gunn diode unit uses a DC power supply for the transmitter, and a separate meter for the receiver. Be careful of the polarity from the power supply to the transmitter. The positive (red) connector of the power supply's output must be hooked to the positive (red) connector of the transmitter's input, or the diode will be destroyed. To adjust this device, place the receiver directly opposite the transmitter, and set the meter to minimum sensitivity. Turn on the power supply, and slowly increase the voltage until the diode begins to generate microwaves. You will notice that a further increase in voltage increases the output power until a plateau voltage is reached. After this point, an increase in voltage does not increase the output power. The transmitter should be operated at the beginning of this plateau. Never exceed 15 Volts DC, as doing so would destroy the Gunn diode.
The microwave receiver consists of a crystal diode which produces a current when aligned parallel to the electric field of the incident microwaves. The diode is not sensitive to microwaves whose electric field is perpendicular to its axis. The current from the diode is read by the horn. Be careful of the polarity between the receiver and the meter. The positive (red) connector of the receiver's output must be hooked to the positive (red) connector of the meter's input, or the meter movement may be destroyed. Never connect the receiver to the power supply, as this will destroy the diode instantly.
PROCEDURE
The microwaves emerging from the transmitter are linearly polarized in the vertical plane, and the receiver is sensitive only to the electric-field component parallel to its axis. Begin by recording the wavelength of the microwaves produced by the transmitter. Stand the transmitter and receiver vertically, with the two horns facing and approximately 30 cm apart from each other, as shown below.
Connect the receiver to the meter, and align the transmitter and receiver horns such that the meter reading is a maximum. Adjust the sensitivity of the meter to read a convenient value (e.g., 100) at the maximum, and take this orientation of the receiver to be \(\theta\) = 0° in Eq. \eqref{eqn_Malus}. Rotate the receiver in 5° increments, and record its reading for angles between 0° and 90° in the “Data” section. Since the meter measures the relative electric-field amplitude of the microwaves, you must square all readings to obtain the relative intensity \(I\). Plot the experimental values of \(I\) as a function of \(\theta\).
Using Eq. \eqref{eqn_Malus}, plot the theoretical values of \(I\) as a function of \(\theta\). Comment on the extent to which your data agrees with or differs from Malus' Law.
Return the receiver to its vertical position. Place the polarization grid between the transmitter and receiver, as shown below.
Rotate the polarization grid until it blocks all incoming microwaves, and note the orientation of the bars with respect to the incident electric field (i.e., either parallel or perpendicular). Explain what is happening. (This is not obvious. It has nothing to do with waves “squeezing between the bars”, but has much to do with the fact that the bars are conductors. You may wish to refer to your data in step 6 for a hint.)
The reflection of microwaves by a full reflector (i.e., an aluminum plate) is measured with the setup shown below.
Place the transmitter approximately 15 cm from the reflector at an angle of incidence \(\theta_{\textrm{i}}\) = 30° (as measured by the protractor on the four-armed base). Vary the angle of the receiver (as measured by the protractor) until the meter reading is a maximum. Record this angle of reflection \(\theta_{\textrm{r}}\). Repeat the procedure for angles of incidence of 45° and 60°, measure the corresponding angles of reflection, and check the validity of the law of reflection.
Arrange the setup for absorption and reflection as shown below.
Place the transmitter and receiver horns facing and approximately 10 cm from each other. Adjust the sensitivity of the meter to read a convenient value (e.g., 100). This is the maximum electric-field amplitude (\(E\,\)) detected by the receiver. Place at least four different materials (two metal, one nonmetal, and lucite) at an angle of 45° with respect to the beam, and record the transmitted amplitude (\(T\,\)) for each material. Rotate the receiver so that it is at a right angle to the transmitter (as measured by the protractor), place the materials at an angle of 45° with respect to the beam, and record the reflected amplitude (\(R\,\)) for each material. Using Eq. \eqref{eqn_fAbsorbed}, calculate the percentage \(f_{\textrm{A}}\) of the incident microwave intensity absorbed by each material.
The wavelength \(\lambda\) of the microwaves can be measured with the Michelson interferometer shown below. (An interferometer is a device that can be used to measure lengths or changes in length with great accuracy by means of interference effects.)
The transmitter and receiver horns should each be approximately 10 cm from the center of the track. Place the two full (aluminum) reflectors at right angles to each other (as measured by the protractor) and at distances \(d_1\) and \(d_2\) from the center of the track. Place the half (lucite) reflector at an angle of 45° with respect to the incident beam. Adjust \(d_1\) (the position of the full reflector opposite the receiver) until the receiver reading is a minimum. Next, adjust \(d_1\) (the position of the full reflector opposite the transmitter) until the receiver reading is a minimum. Then vary \(d_1\) between 15 cm and 40 cm, and record at least 15 values of \(d_1\) for which the receiver output is a minimum. Knowing that the distance between adjacent minima is \(\lambda/2\), calculate \(\lambda\) for each pair of adjacent minima, and determine the average wavelength. Compare this value with the wavelength recorded in step 1.
DATA
Wavelength of microwaves =
Amplitude at \(\theta\) = 0° =
Amplitude at \(\theta\) = 5° =
Amplitude at \(\theta\) = 10° =
Amplitude at \(\theta\) = 15° =
Amplitude at \(\theta\) = 20° =
Amplitude at \(\theta\) = 25° =
Amplitude at \(\theta\) = 30° =
Amplitude at \(\theta\) = 35° =
Amplitude at \(\theta\) = 40° =
Amplitude at \(\theta\) = 45° =
Amplitude at \(\theta\) = 50° =
Amplitude at \(\theta\) = 55° =
Amplitude at \(\theta\) = 60° =
Amplitude at \(\theta\) = 65° =
Amplitude at \(\theta\) = 70° =
Amplitude at \(\theta\) = 75° =
Amplitude at \(\theta\) = 80° =
Amplitude at \(\theta\) = 85° =
Amplitude at \(\theta\) = 90° =
Intensity at \(\theta\) = 0° =
Intensity at \(\theta\) = 5° =
Intensity at \(\theta\) = 10° =
Intensity at \(\theta\) = 15° =
Intensity at \(\theta\) = 20° =
Intensity at \(\theta\) = 25° =
Intensity at \(\theta\) = 30° =
Intensity at \(\theta\) = 35° =
Intensity at \(\theta\) = 40° =
Intensity at \(\theta\) = 45° =
Intensity at \(\theta\) = 50° =
Intensity at \(\theta\) = 55° =
Intensity at \(\theta\) = 60° =
Intensity at \(\theta\) = 65° =
Intensity at \(\theta\) = 70° =
Intensity at \(\theta\) = 75° =
Intensity at \(\theta\) = 80° =
Intensity at \(\theta\) = 85° =
Intensity at \(\theta\) = 90° =
Plot the experimental graph of \(I\) as a function of \(\theta\) using one sheet of graph paper at the end of this workbook. Remember to label the axes and title the graph.
Plot the theoretical graph of \(I\) as a function of \(\theta\) using the same sheet of graph paper. Remember to label the axes and title the graph.
Which orientation of the bars blocks all incoming microwaves? Why?
Angle of reflection for \(\theta_{\textrm{i}}\) = 30° =
Angle of reflection for \(\theta_{\textrm{i}}\) = 45° =
Angle of reflection for \(\theta_{\textrm{i}}\) = 60° =
Maximum electric-field amplitude =
Material 1 =
Material 2 =
Material 3 =
Material 4 =
Transmitted amplitude for material 1 =
Transmitted amplitude for material 2 =
Transmitted amplitude for material 3 =
Transmitted amplitude for material 4 =
Reflected amplitude for material 1 =
Reflected amplitude for material 2 =
Reflected amplitude for material 3 =
Reflected amplitude for material 4 =
Percentage of microwave intensity absorbed by material 1 =
Percentage of microwave intensity absorbed by material 2 =
Percentage of microwave intensity absorbed by material 3 =
Percentage of microwave intensity absorbed by material 4 =
Positions at which receiver output is a minimum =
Wavelength for each pair of adjacent minima =
Average wavelength =
Percentage difference between average wavelength and value recorded in step 1 =
APPARATUS
This lab consists of many short optics experiments. Check over the many pieces of equipment carefully:
Shown in the picture below:
Not shown in the picture above:
If anything is missing, notify your TA. At the end of the lab, you must put everything back in order again, and your TA will check for missing pieces.
REFLECTION AND REFRACTION
When a beam of light enters a transparent material such as glass or water, its overall speed through the material is slowed from \(c\) (3 × 108 m/s) in vacuum by a factor of \(n\) (> 1):
\begin{eqnarray} \textrm{(speed in material)} &=& c/n. \end{eqnarray}
The parameter \(n\) is called the index of refraction, and is generally between 1 and 2 for most transparent materials. Even air has a refractive index slightly greater than 1.
Consider a light beam impinging on the boundary between two transparent materials (e.g., a beam passing from air into glass). By convention, the angle of incidence \(\theta_{\textrm{i}}\) is measured with respect to the normal to the boundary.
In general, the beam will be partially reflected from the boundary at an angle \(\theta_{\textrm{l}}\) with respect to the normal and partially refracted into the material at an angle \(\theta_{\textrm{r}}\) with respect to the normal.
Fermat's Principle, which states that light travels along the path requiring the least time, can be used to derive the laws of reflection and refraction.
Law of Reflection:
\begin{eqnarray} \theta_{\textrm{l}} &=& \theta_{\textrm{i}}. \end{eqnarray}
The angle of reflection \(\theta_{\textrm{l}}\) is equal to the angle of incidence \(\theta_{\textrm{i}}\).
Law of Refraction:
\begin{eqnarray} n_1\sin\theta_{\textrm{i}} &=& n_2\sin\theta_{\textrm{r}}. \end{eqnarray}
This is also called Snell's Law, where \(n_1\) is the refractive index of the material from which light is incident (air in this case), and \(n_2\) is the refractive index of the material to which light is refracted (glass in this case).
THIN LENSES
A thin lens is one whose thickness is small compared to the other characteristic distances (e.g., its focal length). The surfaces of the lens can be either convex or concave, or one surface could be planar. Because of the refractive properties of its surfaces, the lens will either converge or diverge rays that pass through it. A converging lens (such as the first plano-convex lens below) is thicker at its center than at its edges. A diverging lens (such as the second concave meniscus lens below) is thinner at its center than at its edges.
If parallel rays (say, from a distant source) pass through the lens, then a converging lens will bring the rays to an approximate focus at some point behind the lens. The distance between the lens and the focus of parallel rays is called the focal length of the lens.
If the lens is diverging, then parallel rays passing though the lens will spread out, appearing to come from some point in front of the lens. This point is called the virtual focus, and the negative of the distance between the lens and the virtual focus is equal the focal length of the diverging lens.
If an object (say, a lighted upright arrow) is placed near a lens, then the lens will form an image of the arrow at a specific distance from the lens. Let's call the distance between the lens and the object \(d_{\textrm{o}}\), and the distance between the lens and the image \(d_{\textrm{i}}\). Applying the Law of Refraction to the thin lens results in the thin-lens equation, which relates these quantities to the focal length \(f\):
\begin{eqnarray} 1/f &=& 1/d_{\textrm{o}} + 1/d_{\textrm{i}}. \end{eqnarray}
Recall that \(f\) can be positive or negative, depending on whether the lens is converging or diverging, respectively. Once the object distance \(d_{\textrm{o}}\) is chosen, the image distance \(d_{\textrm{i}}\) may turn out to be positive or negative. If \(d_{\textrm{i}}\) is positive, then a real image is formed. A real image focuses on a screen located a distance \(d_{\textrm{i}}\) behind the lens. If \(d_{\textrm{i}}\) is negative, then a virtual image is formed. A virtual image does not focus anywhere, but light emerges from the lens as though it came from an image located a distance \(\mid d_{\textrm{i}} \mid\) in front of the lens. You can see the virtual image by looking back through the lens toward the object. Such an image can be observed when you are looking through a diverging lens. These virtual images looks smaller and more distant.
CURVED MIRRORS
A curved mirror can also converge or diverge light rays that impinge on it. A converging mirror is concave, while a diverging mirror is convex. The mirror equation is identical to the thin-lens equation:
\begin{eqnarray} 1/f &=& 1/d_{\textrm{o}} + 1/d_{\textrm{i}}. \end{eqnarray}
We just need to remember that a real image with a positive image distance \(d_{\textrm{i}}\) will be formed on the same side of the mirror as the incident rays from the object, while a virtual image with a negative image distance \(d_{\textrm{i}}\) will be formed behind the mirror.
PROCEDURE PART 1: REFRACTION AND TOTAL INTERNAL REFLECTION
Place the ray box, label side up, on a white sheet of paper on the table. Plug in its transformer. Adjust the box so that one white ray is showing.
Position the rhombus as shown in the figure. The triangular end of the rhombus is used as a prism in this experiment. Keep the ray near the point of the rhombus for the maximum transmission of light. Notice that a refracted ray emerges from the second surface, and a reflected ray continues in the acrylic of the rhombus.
The incident ray is bent once as it enters the acrylic of the rhombus, and again as it exits the rhombus. Vary the angle of incidence. Does the exiting ray bend toward or away from the normal? (Physicists and opticians measure the angles of the rays with respect to the normal, a line perpendicular to the surface.)
Does the exiting ray bend toward or away from the normal?
Pick an angle of incidence for which the exiting ray is well bent, and trace neatly the internal and exiting rays on the top half of the paper underneath. Also trace the rhombus-air interfaces, clearly marking the side corresponding to the rhombus and that corresponding to air. You can simply mark the ends of the rays and use a ruler to extend the rays. Use the protractor to construct the normal to the interface and measure the angles of the two rays with respect to the normal. With these angles, use Snell's Law to find the refractive index of the acrylic of the rhombus. (Use \(n=1\) for air.)
Angle of ray in acrylic =
Angle of ray in air =
Refractive index (\(n\)) of acrylic =
Show your calculation of \(n\) below:
Total internal reflection: Rotate the rhombus until the exiting ray travels parallel to the surface (separating into colors), and then rotate the rhombus slightly farther. Now there is no refracted ray; the light is totally internally reflected from the inner surface. Total internal reflection occurs only beyond a certain “critical angle” \(\theta_{\textrm{c}}\), the angle at which the exiting refracted ray travels parallel to the surface. Rotate the rhombus again, and notice how the reflected ray becomes brighter as you approach and reach the critical angle. When there is both a refracted ray and a reflected ray, the incident light energy is divided between these rays. However, when there is no refracted ray, all of the incident energy goes into the reflected ray (minus any absorption losses in the acrylic).
Adjust the rhombus exactly to the critical angle, and trace neatly the ray in the acrylic and the refracting surface on the bottom half of the paper. Construct the normal to the surface, and measure the critical angle of the ray. (Again, all angles are measured with respect to the normal.) According to the textbook, the sine of the critical angle is
\begin{eqnarray} \sin\theta_{\textrm{c}} &=& 1/n. \end{eqnarray}
Calculate \(n\) from this relation, and compare it to the \(n\) determined in step 4.
Measured critical angle \(\theta_{\textrm{c}}\) =
Refractive index (\(n\)) determined from critical angle =
Refractive index (\(n\)) determined from Snell's Law (copy from step 4) =
Adjust the rhombus until the angle of the exiting ray is as large as possible (but less than the critical angle) and still clearly visible, and the exiting ray separates into colors. This phenomenon is called dispersion and illustrates the refraction of different colors at various angles. Which color is refracted at the largest angle, and which color is refracted at the smallest angle?
Color refracted at largest angle =
Color refracted at smallest angle =
PROCEDURE PART 2: REFLECTION
As in the preceding section, the ray box should be on a white sheet of paper, label side up, with one white ray showing.
Place the triangular-shaped mirror piece on the paper, and position the plane surface so that both the incident and reflected rays are clearly seen.
By turning the mirror piece, vary the angle of incidence while observing how the angle of reflection changes. What is the relation between the angle of incidence and the angle of reflection?
Relation:
PROCEDURE PART 3: CONVERGENCE AND DIVERGENCE OF RAYS
Either a mirror or a lens can converge or diverge parallel rays. The triangular mirror piece has a concave and a convex side, and there is a section of a double convex lens and a double concave lens. Adjust the ray box so that it makes five parallel white rays.
The concave mirror and the double convex lens (shown above) both converge the parallel rays to an approximate focal point. The distance between the lens or mirror surface and the focal point of parallel rays is the focal length of the mirror or lens. Measure the focal lengths of the concave mirror and convex lens in centimeters, and enter them in the table below.
The convex mirror and the double concave lens both diverge the parallel rays. They have negative focal lengths, and the magnitude of the focal length is equal to the distance between the optical element and the point from which the rays appear to diverge. Using the convex mirror and the double concave lens (one at a time), sketch the mirror or lens surface in position, and trace the diverging rays on the white paper. Remove the mirror or lens, and continue tracing the rays back to the virtual focus using a ruler. Enter the (negative) focal lengths (in centimeters) of these optical elements in the table above.
PROCEDURE PART 4: IMAGE FORMATION AND FOCAL LENGTH OF A LENS
Place the 200-mm lens and the screen on the optical-bench track. Do not put the light source on yet.
Focus a distant light source (such as a window, the trees outside the window, or a light at the other end of the room) on the screen. A distant source is effectively at infinity, the rays from the source are parallel, and the lens converges the rays to an image at the focal point. Measure the distance between the lens and the screen, and compare this distance to the stated focal length. (You may read positions off the optical bench scale and subtract them to find distances.)
Notice that the image is inverted. This is similar to how your eye lens forms an inverted image of the outside world on your retina.
Now mount the light source with the circles and arrows side facing the lens and screen. You need to unplug the power cord of the light box, and then replug it when the box is mounted. (There are two ways to mount the light source in the bracket. Notice the two holes in the bracket for the detent buttons on either side. For one way, the offset of the bracket arm below permits reading the position of the light source directly from the scale; for the other way, you would need to correct for the setback of the light source.)
Adjust the position of the lens until the image of the light source is focused sharply on the screen. Read the distances \(d_{\textrm{o}}\) and \(d_{\textrm{i}}\) in the figure off the scale, and calculate the focal length of the lens from the thin-lens equation:
\begin{eqnarray} 1/f &=& 1/d_{\textrm{o}} + 1/d_{\textrm{i}}. \end{eqnarray}
Enter the data below:
\(d_{\textrm{o}}\) =
\(d_{\textrm{i}}\) =
\(1/f\) =
\(f\) (calculated) =
\(f\) (theoretical) =
Here \(f\) (calculated) is the value obtained from the thin-lens equation, and \(f\) (theoretical) is the value read off the lens. These two focal lengths should, of course, agree approximately.
PROCEDURE PART 5: TWO-LENS SYSTEMS
Place the light source at 110 cm and the screen at 60 cm on the optical-bench scale. Place the +100 mm lens between the light source and the screen. You will find it possible to obtain a sharp image on the screen with the lens at about 72 cm. Is the image upright or inverted? Adjust for the exact focus, measure the object and image distances (read positions off the scale and subtract them), and compare the theoretical focal length with that obtained from the thin-lens equation.
\(d_{\textrm{o}}\) =
\(d_{\textrm{i}}\) =
\(1/f\) =
\(f\) (calculated) =
\(f\) (theoretical) =
Is the image upright or inverted?
The image formed at 60 cm can serve as the object for a second lens. Move the screen back, and place the +25 mm lens at 55 cm on the scale. Where does the image focus? Is it upright or inverted? Measure the object and image distances for the +25 mm lens, and compare the theoretical focal length with that obtained from the thin-lens equation.
\(d_{\textrm{o}}\) =
\(d_{\textrm{i}}\) =
\(1/f\) =
\(f\) (calculated) =
\(f\) (theoretical) =
Is the image upright or inverted?
If a second lens is placed inside the focal point of the first lens, the image of the first lens serves as a virtual object for the second lens. Place the +200 mm lens at 68 cm on the scale. The object distance is the negative of the distance between the lens and the point at which the image would have formed: namely, −8.0 cm if the first image were at 60 cm and the second lens were at 68 cm. Where does the image focus now? Is it upright or inverted? Compare the theoretical focal length with that obtained from the thin-lens equation.
\(d_{\textrm{o}}\) =
\(d_{\textrm{i}}\) =
\(1/f\) =
\(f\) (calculated) =
\(f\) (theoretical) =
Is the image upright or inverted?
Repeat step 3, placing the diverging −25 mm lens at 68 cm on the scale. This strongly diverging lens bends the rays from the +100 mm lens outward so that they diverge and never come to a focus beyond the lens. Instead, look through the two lenses back to the source. You will see the virtual image at a distance. Is it upright or inverted? Calculate the image distance of the −25 mm lens with its virtual object. The result comes out negative. Look through the lenses again. Does the image distance seem reasonable?
\(d_{\textrm{o}}\) =
\(d_{\textrm{i}}\) =
Does image distance seem reasonable?
Is the image upright or inverted?
PROCEDURE PART 6: SIMPLE TELESCOPES
You have four lenses in holders; the lenses have focal lengths of +200 mm, +100 mm, +25 mm, and −25 mm. Carry these lenses over to a window so that you can look out at distant objects (such as a building across the quadrangle). DO NOT LOOK AT THE SUN WITH YOUR TELESCOPE ARRANGEMENTS. PERMANENT EYE DAMAGE MAY RESULT. To make a telescope, hold one of the short focal-length lenses near your eye and one of the longer focal-length lenses out with your arm, so that you look through both lenses in series. Adjust the position of the second lens (the objective) until a distant object is focused. The lens nearest your eye is called the eye lens, and the one farther out is called the objective.
Galilean telescope: Use the negative focal-length lens (\(f\) = −25 mm) as the eye lens and the +100 mm or +200 mm lens as the objective, and focus a distant object. Notice that the field of view is small and the image is distorted. Nevertheless, Galilei used an optical arrangement similar to this to discover the moons of Jupiter, the phases of Venus, sunspots, and many other heavenly wonders.
Astronomical telescope: Use the +25 mm lens as the eye lens and the +100 mm or +200 mm lens as the objective, and focus a distant object. Notice that the field of view is now larger and the image is sharper, although the image is inverted. You can also try the +100 mm lens as the eye lens and the +200 mm lens as the objective.
PROCEDURE PART 7: MEASURING THE POWER OF AN ASTRONOMICAL TELESCOPE
Use the +25 mm lens as the eye lens and the +100 mm lens as the objective. Place the lenses near one end of the optical bench and the screen at the other end, as shown below. Tape a piece of graph paper to the screen. (Graph paper and tape are in the lab room.)
Look through the eye lens, and focus the image of the graph paper by moving the objective.
(This procedure is a bit complex. Try your best and do not waste a lot of time on it.) Eliminate parallax by moving the eye lens until the image is in the same plane as the object (the screen). To observe the parallax, open both eyes and look through the lens at the image with one eye, while looking around the edge of the lens directly at the object with the other eye. Refer to the figures below. The lines of the image (solid lines in the figure below) will be superimposed on the lines of the object (dotted lines in the figure below). Move your head back and forth, and up and down. As you move your head, the lines of the image will move relative to the lines of the object due to parallax. To eliminate parallax, adjust the eye lens until the image lines do not move relative to the object lines when you move your head. When there is no parallax, the lines in the center of the lens appear to be stuck to the object lines. (Even when there is no parallax, the lines may appear to move near the edge of the lens because of lens aberrations.)
Measure the magnification of this telescope by counting the number of squares in the object that lie along a side of one square of the image. To do this, you must view the image through the telescope with one eye, while looking directly at the object with the other eye. Record the observed magnification in step 5.
The theoretical magnification for objects at infinity is equal to the ratio of the focal lengths. Record and compare the theoretical and observed magnifications below.
Observed magnification =
Theoretical magnification =
ADDITIONAL CREDIT PART 1: MEASURING A GLASSES PRESCRIPTION (3 mills)
The inverse of the focal length of a lens, \(P = 1/f\), is called the power of the lens. The units of power are inverse meters which are renamed diopters, a unit commonly used by optometrists and opticians. The larger the power, the more strongly the lens converges rays. You can show that when two thin lenses are placed close together (so that the distance between them is much less than the focal lengths), the power \(P_{\textrm{T}}\) of the combined lenses is the sum of the powers \(P_1\) and \(P_2\) of the individual lenses:
\begin{eqnarray} P_{\textrm{T}} &=& P_1 + P_2 \end{eqnarray}
or
\begin{eqnarray} 1/f_{\textrm{T}} &=& 1/f_1 + 1/f_2. \end{eqnarray}
The closest distance at which you can focus your eyes clearly (when you are exerting maximum muscle tension on your eye lens) is called your near point. The farthest distance at which you can focus your eyes clearly (when your focusing muscles are relaxed) is called your far point. Ideally, your far point is at infinity, and your near point is at least as small as 25 cm so you can read easily. If you are nearsighted, then your far point is at some finite distance; you cannot focus distant objects clearly. If you are farsighted, then your far point is “beyond infinity”, so to speak, so that you need to exert eye-lens muscle tension even to focus distant objects. As you grow older, your power of accommodation (i.e., your ability to change the focal length of your eye lens) weakens and your near point moves out, so that you must have corrective lenses to focus on close objects, such as for reading. Thus, you notice older persons wearing reading glasses.
You may be wearing glasses, contact lenses, or have had laser eye surgery to correct your vision — or you may be lucky and have “perfect” vision without correction. In any case, use a meter stick (or other ruler) and the card with fine print to measure your near point (with correction, if any) as in the illustration below. The purpose of laying the meter stick on the table is to avoid poking it toward your eye.
Move the card in to the closest distance that you can focus clearly.
Near-point distance (corrected) =
If you are nearsighted and wearing glasses, take off your glasses and measure your far point. (If you are wearing contacts, you may remove a contact and try this, but the step is optional.)
Far-point distance (uncorrected) =
If you or your lab partner are nearsighted and wearing glasses, determine your glasses prescription as instructed below. If neither you nor your partner is nearsighted and wearing glasses, use the (uncorrected) data for Dr. Art Huffman: far point = 20 cm, near point = 18 cm. (Yes, his vision is that bad!)
If the eye is nearsighted, we want to put a diverging lens in front of it, which will shift the uncorrected far point to infinity. We can use the formula above to find the focal length of the glasses-eye combination. Let the (uncorrected) far-point distance be \(d\), the eye-to-retina distance be \(i\), the focal length of the eye lens while relaxed be \(f_{\textrm{f}}\), and the focal length of the glasses be \(f_{\textrm{g}}\):
\begin{eqnarray} \textrm{Without glasses:} &\hspace{10pt}& 1/d + 1/i = 1/f_{\textrm{f}} \\ \textrm{With glasses:} &\hspace{10pt}& 1/\infty + 1/i = 1/f_{\textrm{f}} + 1/f_{\textrm{g}}. \end{eqnarray}
Subtracting the first equation from the second gives the power \(P_{\textrm{g}}\) of the glasses:
\begin{eqnarray} P_{\textrm{g}} &=& 1/f_{\textrm{g}} = -1/d. \end{eqnarray}
Compute your glasses prescription (or Art's) in diopters:
ADDITIONAL CREDIT PART 2: MEASURING THE FOCAL LENGTH OF A DIVERGING LENS (2 mills)
Devise a way, using your optical bench, to measure the focal length of a diverging lens. Then measure the focal length of your glasses as in Additional Credit Part 1, or measure the focal length of one of the unknown lenses supplied in the lab.
Sketch your plan for measuring the focal length of a diverging lens below, and report the measured power of the glasses or of the unknown lens.
APPARATUS
INTRODUCTION
The energy quantization of electromagnetic radiation in general, and of light in particular, is expressed in the famous relation
\begin{eqnarray} E &=& hf, \label{eqn_1} \end{eqnarray}
where \(E\) is the energy of the radiation, \(f\) is its frequency, and \(h\) is Planck's constant (6.63×10-34 Js). The notion of light quantization was first introduced by Planck. Its validity is based on solid experimental evidence, most notably the photoelectric effect. The basic physical process underlying this effect is the emission of electrons in metals exposed to light. There are four aspects of photoelectron emission which conflict with the classical view that the instantaneous intensity of electromagnetic radiation is given by the Poynting vector \(\textbf{S}\):
\begin{eqnarray} \textbf{S} &=& (\textbf{E}\times\textbf{B})/\mu_0, \label{eqn_2} \end{eqnarray}
with \(\textbf{E}\) and \(\textbf{B}\) the electric and magnetic fields of the radiation, respectively, and μ0 (4π×10-7 Tm/A) the permeability of free space. Specifically:
No photoelectrons are emitted from the metal when the incident light is below a minimum frequency, regardless of its intensity. (The value of the minimum frequency is unique to each metal.)
Photoelectrons are emitted from the metal when the incident light is above a threshold frequency. The kinetic energy of the emitted photoelectrons increases with the frequency of the light.
The number of emitted photoelectrons increases with the intensity of the incident light. However, the kinetic energy of these electrons is independent of the light intensity.
Photoemission is effectively instantaneous.
THEORY
Consider the conduction electrons in a metal to be bound in a well-defined potential. The energy required to release an electron is called the work function \(W_0\) of the metal. In the classical model, a photoelectron could be released if the incident light had sufficient intensity. However, Eq. \eqref{eqn_1} requires that the light exceed a threshold frequency \(f_{\textrm{t}}\) for an electron to be emitted. If \(f > f_{\textrm{t}}\), then a single light quantum (called a photon) of energy \(E = hf\) is sufficient to liberate an electron, and any residual energy carried by the photon is converted into the kinetic energy of the electron. Thus, from energy conservation, \(E = W_0 + K\), or
\begin{eqnarray} K &=& (1/2)mv^2 = E - W_0 = hf - W_0. \label{eqn_3} \end{eqnarray}
When the incident light intensity is increased, more photons are available for the release of electrons, and the magnitude of the photoelectric current increases. From Eq. \eqref{eqn_3}, we see that the kinetic energy of the electrons is independent of the light intensity and depends only on the frequency.
The photoelectric current in a typical setup is extremely small, and making a precise measurement is difficult. Normally the electrons will reach the anode of the photodiode, and their number can be measured from the (minute) anode current. However, we can apply a reverse voltage to the anode; this reverse voltage repels the electrons and prevents them from reaching the anode. The minimum required voltage is called the stopping potential \(V_{\textrm{s}}\), and the “stopping energy” of each electron is therefore \(eV_{\textrm{s}}\). Thus,
\begin{eqnarray} eV_{\textrm{s}} &=& hf - W_0, \label{eqn_4} \end{eqnarray}
or
\begin{eqnarray} V_{\textrm{s}} &=& (h/e)f - W_0/e. \label{eqn_5} \end{eqnarray}
Eq. \eqref{eqn_5} shows a linear relationship between the stopping potential \(V_{\textrm{s}}\) and the light frequency \(f\), with slope \(h/e\) and vertical intercept \(-W_0/e\). If the value of the electron charge \(e\) is known, then this equation provides a good method for determining Planck's constant \(h\). In this experiment, we will measure the stopping potential with modern electronics.
THE PHOTODIODE AND ITS READOUT
The central element of the apparatus is the photodiode tube. The diode has a window which allows light to enter, and the cathode is a clean metal surface. To prevent the collision of electrons with air molecules, the diode tube is evacuated.
The photodiode and its associated electronics have a small “capacitance” and develop a voltage as they become charged by the emitted electrons. When the voltage across this “capacitor” reaches the stopping potential of the cathode, the voltage difference between the cathode and anode (which is equal to the stopping potential) stabilizes.
To measure the stopping potential, we use a very sensitive amplifier which has an input impedance larger than 1013 ohms. The amplifier enables us to investigate the minuscule number of photoelectrons that are produced.
It would take considerable time to discharge the anode at the completion of a measurement by the usual high-leakage resistance of the circuit components, as the input impedance of the amplifier is very high. To speed up this process, a shorting switch is provided; it is labeled “Push to Zero”. The amplifier output will not stay at 0 volts very long after the switch is released. However, the anode output does stabilize once the photoelectrons charge it up.
There are two 9-volt batteries already installed in the photodiode housing. To check the batteries, you can use a voltmeter to measure the voltage between the output ground terminal and each battery test terminal. The battery test points are located on the side panel. You should replace the batteries if the voltage is less than 6 volts.
THE MONOCHROMATIC LIGHT BEAMS
This experiment requires the use of several different monochromatic light beams, which can be obtained from the spectral lines that make up the radiation produced by excited mercury atoms. The light is formed by an electrical discharge in a thin glass tube containing mercury vapor, and harmful ultraviolet components are filtered out by the glass envelope. Mercury light has five narrow spectral lines in the visible region — yellow, green, blue, violet, and ultraviolet — which can be separated spatially by the process of diffraction. For this purpose, we use a high-quality diffraction grating with 6000 lines per centimeter. The desired wavelength is selected with the aid of a collimator, while the intensity can be varied with a set of neutral density filters. A color filter at the entrance of the photodiode is used to minimize room light.
The equipment consists of a mercury vapor light housed in a sturdy metal box, which also holds the transformer for the high voltage. The transformer is fed by a 115-volt power source from an ordinary wall outlet. In order to prevent the possibility of getting an electric shock from the high voltage, do not remove the cover from the unit when it is plugged in.
To facilitate mounting of the filters, the light box is equipped with rails on the front panel. The optical components include a fixed slit (called a light aperture) which is mounted over the output hole in the front cover of the light box. A lens focuses the aperture on the photodiode window. The diffraction grating is mounted on the same frame that holds the lens, which simplifies the setup somewhat. A “blazed” grating, which has a preferred orientation for maximal light transmission and is not fully symmetric, is used. Turn the grating around to verify that you have the optimal orientation.
The variable transmission filter consists of computer-generated patterns of dots and lines that vary the intensity of the incident light. The relative transmission percentages are 100%, 80%, 60%, 40%, and 20%.
INITIAL SETUP
Your apparatus should be set up approximately like the figure above. Turn on the mercury lamp using the switch on the back of the light box. Swing the \(h/e\) apparatus box around on its arm, and you should see at various positions, yellow green, and several blue spectral lines on its front reflective mask. Notice that on one side of the imaginary “front-on” perpendicular line from the mercury lamp, the spectral lines are brighter than the similar lines from the other side. This is because the grating is “blazed”. In you experiments, use the first order spectrum on the side with the brighter lines.
Your apparatus should already be approximately aligned from previous experiments, but make the following alignment checks. Ask you TA for assistance if necessary.
Check the alignment of the mercury source and the aperture by looking at the light shining on the back of the grating. If necessary, adjust the back plate of the light-aperture assembly by loosening the two retaining screws and moving the plate to the left or right until the light shines directly on the center of the grating.
With the bright colored lines on the front reflective mask, adjust the lens/grating assembly on the mercury lamp light box until the lines are focused as sharply as possible.
Roll the round light shield (between the white screen and the photodiode housing) out of the way to view the photodiode window inside the housing. The phototube has a small square window for light to enter. When a spectral line is centered on the front mask, it should also be centered on this window. If not, rotate the housing until the image of the aperture is centered on the window, and fasten the housing. Return the round shield back into position to block stray light.
Connect the digital voltmeter (DVM) to the “Output” terminals of the photodiode. Select the 2 V or 20 V range on the meter.
Press the “Push to Zero” button on the side panel of the photodiode housing to short out any accumulated charge on the electronics. Note that the output will shift in the absence of light on the photodiode.
Record the photodiode output voltage on the DVM. This voltage is a direct measure of the stopping potential.
Use the green and yellow filters for the green and yellow mercury light. These filters block higher frequencies and eliminate ambient room light. In higher diffraction orders, they also block the ultraviolet light that falls on top of the yellow and green lines.
PROCEDURE PART 1: DEPENDENCE OF THE STOPPING POTENTIAL ON THE INTENSITY OF LIGHT
Adjust the angle of the photodiode-housing assembly so that the green line falls on the window of the photodiode.
Install the green filter and the round light shield.
Install the variable transmission filter on the collimator over the green filter such that the light passes through the section marked 100%. Record the photodiode output voltage reading on the DVM. Also determine the approximate recharge time after the discharge button has been pressed and released.
Repeat steps 1 – 3 for the other four transmission percentages, as well as for the ultraviolet light in second order.
Plot a graph of the stopping potential as a function of intensity.
PROCEDURE PART 2: DEPENDENCE OF THE STOPPING POTENTIAL ON THE FREQUENCY OF LIGHT
You can see five colors in the mercury light spectrum. The diffraction grating has two usable orders for deflection on one side of the center.
Adjust the photodiode-housing assembly so that only one color from the first-order diffraction pattern on one side of the center falls on the collimator.
For each color in the first order, record the photodiode output voltage reading on the DVM.
For each color in the second order, record the photodiode output voltage reading on the DVM.
Plot a graph of the stopping potential as a function of frequency, and determine the slope and the \(y\)-intercept of the graph. From this data, calculate \(W_0\) and \(h\). Compare this value of \(h\) with that provided in the “Introduction” section of this experiment.
DATA
Procedure Part 1:
Photodiode output voltage reading for 100% transmission =
Approximate recharge time for 100% transmission =
Photodiode output voltage reading for 80% transmission =
Approximate recharge time for 80% transmission =
Photodiode output voltage reading for 60% transmission =
Approximate recharge time for 60% transmission =
Photodiode output voltage reading for 40% transmission =
Approximate recharge time for 40% transmission =
Photodiode output voltage reading for 20% transmission =
Approximate recharge time for 20% transmission =
Photodiode output voltage reading for ultraviolet light =
Approximate recharge time for ultraviolet light =
Plot the graph of stopping potential as a function of intensity using one sheet of graph paper at the end of this workbook. Remember to label the axes and title the graph.
Procedure Part 2:
First-order diffraction pattern on one side of the center:
Photodiode output voltage reading for yellow light =
Photodiode output voltage reading for green light =
Photodiode output voltage reading for blue light =
Photodiode output voltage reading for violet light =
Photodiode output voltage reading for ultraviolet light =
Second-order diffraction pattern on the other side of the center:
Photodiode output voltage reading for yellow light =
Photodiode output voltage reading for green light =
Photodiode output voltage reading for blue light =
Photodiode output voltage reading for violet light =
Photodiode output voltage reading for ultraviolet light =
Plot the graph of stopping potential as a function of frequency using one sheet of graph paper at the end of this workbook. Remember to label the axes and title the graph.
Slope of graph =
\(y\)-intercept of graph =
\(W_0\) =
\(h\) =
Percentage difference between experimental and accepted values of \(h\) =
Links:
[1] https://demoweb.physics.ucla.edu/sites/default/files/Lab%20Hazard%20Awareness%20Information.pdf
[2] https://demoweb.physics.ucla.edu/sites/default/files/LASER%20LITE.pdf
[3] https://demoweb.physics.ucla.edu/sites/default/files/Physics6C_Exp4.pdf
[4] https://demoweb.physics.ucla.edu/sites/default/files/Lab%20Hazard%20Awareness%20Information_0.pdf
[5] https://demoweb.physics.ucla.edu/sites/default/files/LASER%20LITE_0.pdf
[6] https://demoweb.physics.ucla.edu/sites/default/files/Physics6C_Exp5.pdf
[7] https://demoweb.physics.ucla.edu/sites/default/files/Physics6C_Exp7.pdf