User:Michael R Phillips/Notebook/Physics 307L/2008/11/26

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 * style="background-color: #339999"|[[Image:PhotonIdentityCartoon.gif|145px]] Planck's Constant Lab
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Introduction & Safety
For this lab, we will be measuring a value for Planck's constant (h) by relating it to photoelectric effects caused by a Mercury lamp's light incident upon a cathode. There a few safety concerns that we will need to be careful about. These are high voltage, hot Mercury tube, and risk of breaking the tube or lens/grating. Most of these hazards have already been dealt with in previous labs (see Balmer Series) so we were already prepared for them. The only other important thing to note as far as safety is, as Aram pointed out, that we do not want to allow too strong of an intensity of light into our h/e apparatus because it could be damaging. To account for this, we will make sure to attach a filter which will simply decrease the intensity of light entering the apparatus.

Setup & Equipment
Equipment
 * Wavetek 85XT Multimeter
 * Pasco Scientific OS-9286 Hg Light Source
 * Pasco Scientific AP-9368 h/e Apparatus
 * Stephen Martinez's Armitron Watch (used as a timer)

Setup

The lab was setup already, but we will still supply a brief description. We have a Mercury light source outfitted with a converging lens and diffraction grating pointed towards a "h/e" apparatus. This h/e apparatus is just a small circuit contained in a box. This circuit is basically just a piece of metal (the "cathode") which will receive photons from the Mercury lamp connected in series to a battery and an output (and ground). The battery is there so that we will have a voltage source that we can vary to measure the stopping potential from the output. The ground just resets the circuit so that the potential is zero again just before the cathode.

Before really getting started, we had to make sure our battery in the h/e apparatus still had enough voltage, so we connected the multimeter in parallel with the "Battery Test" connectors on the apparatus. We know, from the manual and from speaking with Aram, that the apparatus has two 9-volt batteries inside, giving a maximum potential of 18V if the batteries were new. The apparatus shows the minimum values, indicating ±6V which gives us a minimum of 12V that could be read through our multimeter. We actually measured our maximum potential to be

$$V_{max}=(16.091 \pm .001)V$$

which is between the maximum possible (i.e. new batteries) and minimum possible (i.e. old batteries, not enough to stop photon energy) potentials.

Now that we know that we have good enough batteries to perform the experiment, we can start really setting things up. We begin by connecting our multimeter to the output connections on our h/e apparatus. This will display the exact stopping potential for the incident light. Next, we adjusted the distance between the light source and the lens/grating assembly, which slides on a pair of rods, until the light from the source was focused at the distance of the white reflective plate just in front of the apparatus input. We then rotated the apparatus until the focused light was centered (and focused) on the photodiode behind the light shield and reflective plate. After discharging the circuit to eliminate ambient voltage, we are ready to begin with the real procedure.

Experiment 1
Experiment 1 consists of measurements that prove that the quantum theory of light is true and that the classical model for energy in light (which relates to intensity instead of frequency) is wrong.

The classical theory of light states that

$$V_{stop} \propto I_0$$

which asserts that the stopping potential (and thus the kinetic energy of the photons) is proportional to the intensity of the incident light.

The quantum theory of light, however, states that

$$V_{stop} = \frac{E - W_0}{e} = \frac{h \cdot \nu - W_0}{e}$$

which asserts that the stopping potential is related directly to the frequency of the light. The W0 term indicates the work function of the metal that the light is incident upon.

Therefore, to show that classical thought is incorrect, all we need to do is show that the stopping potential is in fact proportional to the frequency, not the intensity, of the incident light. We do expect, however, the intensity of the light to affect certain things (like current, for example) so we will measure the "recharge time" which is the time for the potential to rise back up to the true potential after clearing out the energy in the circuit.

The following are tables for a few given colors and their corresponding frequencies emitted by Mercury that show our measurements for the stopping potentials with relation to recharge time and relative intensity.

For Ultraviolet (ν = 8.2·1014 Hz):

For Violet (ν = 7.41·1014 Hz):

For Yellow (ν = 5.19·1014 Hz):

Experiment 2
For the second part of this lab, we will measure the value for Planck's constant, h, by using the conclusion from Experiment 1. We will use the same relation above for the stopping potential to determine this constant, since it is simply the proportionality constant for the transition from kinetic energy to frequency.

$$V_{stop} = \frac{h \cdot \nu - W_0}{e}$$

In order to get good enough data to get h within any kind of uncertainty, we need to take many data sets. Therefore, we will record data for all colors. Also, we will want to record data for both the first and second order maxima.

First Order:

Second Order:

Analysis & Conclusions
For analysis, we will first plot our data. For Experiment 1, we will plot the relative intensity percentage with both the stopping potentials and the recharge time for all the colors we gathered data for. This will not be examined closely, but will give us an idea of the trends these values form. For Experiment 2, we will plot the stopping potentials for all colors, both first and second order, vs the frequencies (dictated by the colors). We will then use the equation relating stopping potential to frequency and Planck's Constant (h), as stated above, to determine a value for Planck's Constant.

$$V_{stop}(\nu) = \frac{h \cdot \nu}{e} - \frac{W_0}{e} = m \cdot \nu - b$$

This value is embedded in the slope (m) for the extracted line (using Excel's LINEST aka Least Squares Fitting) and is related directly to the known value for the charge of an electron. We will also be able to extract what the value of our work function is because it is just related directly to the y-intercept (b) for our stopping potential function (again with an electron charge factor).

Along with this analysis, the lab manual suggests that we comment on some of the effects relating to the experiments in this lab:

Experiment 1
As we changed the amount of intensity allowed into the apparatus, we found that the stopping potential decreased by very small amounts (on the order of 0.1 Volts). This means that electrons in the circuit formed from the photoelectric effect had slightly less kinetic energy than ones formed with 100% intensity. The effect this filter had on recharge time was much more noticeable, as seen in our plots. The trend shows that recharge times for low intensities were much longer than those for high intensities. It is important to note that our method of measuring this recharge time (which was just using Stephen's watch's timer function) was highly inaccurate, but this does not really affect our data. We had large errors in this part of the experiment, which is clear by our semi-scattered data points on the plot, but the trend is very clear and it's only the trend that is important.

Different colors of light showed very significant changes in the stopping potential, as shown in our data. This clearly means that the kinetic energy of photoelectric electrons is highly dependent on frequency and allows us to make the assumption

$$E \propto \nu$$

This assumption (or result, rather) leads us to believe that the photoelectric effect is a result of quantum (particle) effects of light and not wave effects. This is because waves are generally said to store energy as amplitude, but our data shows that there is little or no dependence on amplitude (intensity). Our results do, however, suggest the dependence on frequency which is predicted with the quantum theory of light.

There is still a slight issue with our results, however, as far as the small variance (and common trend) in the stopping potentials, and therefore energies, while varying the incident intensity of the light. So why does this happen? This requires some insight on how our apparatus actually operates. The main idea is the way we are measuring the stopping potential: we are actually passing the current through a very high impedance with a one-to-one input-output then into a capacitor. The potential is then measured as the potential across the capacitor (this is what the "recharge time" comes from). Since there is a smaller number of electrons being ejected by the light at lower intensities, it makes sense for the recharge time to be longer (since the capacitor will take longer to charge with fewer electrons entering the anode). So why do low intensities affect the voltage as well? Since there are fewer electrons being ejected, there is less current in the circuit. This wouldn't normally affect the system, but since our impedance is less than infinity, the voltmeter actually steals just a little current. With high currents, this affect is quite small, but with lower currents (lower intensities) the effect becomes somewhat significant relative to the incoming current. Mathematically speaking, the current sucked through the voltmeter is

$$i_{loss} \approx 0$$

for high incident intensity but

$$i_{loss} \not\approx 0 $$

for low incident intensity. Or more generally

$$i_{loss} \propto \frac{1}{I_{rel}}$$

Experiment 2
There is no new discussion for Experiment 2, but there are some different results. The method of obtaining a value for Planck's Constant has already been explained, but we also need to discuss how we compare this with the accepted value.

$$h_{acc} = 6.626\,068\,96(33) \times 10^{-34}~\mathrm{J}\cdot\mathrm{s}$$

For comparison, we will use the equation

$$%error = \frac{|h_{acc}-h_{exp}|}{h_{acc}}$$

which will tell us the percent difference between the accepted and our experimental values.

Excel Files

Here are the files that show our work in Excel to calculate our values. Experiment 1 and the second order part of Experiment 2 are in their own worksheets in [[Media:Planck.xlsx | Planck.xlsx]] and the first order part of Experiment 2 is in [[Media:First Order.xlsx | First Order.xlsx]].


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