Download The Photoelectric Effect

Physics 18-L
Modern Physics Lab
The Photoelectric Effect:
Measuring h/e
MacDonald, et al.
2.4.04
Abstract
The photoelectric effect is the emission of electrons from a metal when illuminated by a source of
electromagnetic radiation. This was one of the first phenomenon modeled using quantized light
(which earned Einstein a Nobel Prize in 1921) which prompted physicists to embrace the notion
of wave-particle duality and helped develop quantum mechanics. Along with the Compton
Effect, atomic spectra, and the black-body spectra, it exposed some grave inadequacies of
classical physics which inevitably led to the paradigm shift we now recognize as modern physics.
Our aim here is to study the properties of a photoelectric system. Specifically, we determine
the ratio h/e; Planck’s constant (a ubiquitous quantity of paramount importance in quantum
mechanics) to the charge of the electron (long believed to be the fundamental charge in nature).
In addition, we obtain the value of the work function (φ) of the metal in our system.
Method
The photoelectric system is in vacuum and consists of a two metal plates separated by a small
distance; one (cathode) is connected to a voltage source and faces a mercury lamp whose
emission lines supply different light frequencies while the other plate (anode) is grounded. An
ammeter is set to measure current between anode and voltage source. When the cathode is
irradiated it emits electrons with a maximum kinetic energy (KEmax ) given by
KEmax = hf − φ.
The emitted electrons are attracted to the anode and register a current on the ammeter. Now
KEmax can be determined by increasing the potential on the cathode until the current is nulled.
The voltage at which this occurs is the stopping voltage (Vs ), and is related to the maximum
kinetic energy via KEmax = eVs . Using this with the equation above yields an expression in
terms of our observables, to wit,
KEmax = eVs = hf − φ
φ
h
⇒ Vs = f − .
e
e
As the frequency varies, we expect a linear response for the stopping potential with a frequency
cutoff due to the metal’s work function. The cutoff occurs when the right hand side of the last
equation vanishes (meaning that Vs = 0 and thus the electrons have no kinetic energy). In
terms of the work function, the cutoff is given by:
φ
h
φ
h
fcutoff − = 0 ⇒ fcutoff =
e
e
e¢
e¢
φ
⇒ fcutoff = .
h
Noting how the stopping potential responds to a varying frequency reveals the important features predicted by the model.
Data, Analysis, and Calculations
Figure 1 shows the behavior of the stopping voltage as the frequency varies. The slope of
the best fit line is the ratio h/e and the y-intercept is the fraction −φ/e. The former ratio is
3.90 × 10−15 J·s/C (or alternatively V·s) and the later is −1.37J/C (or V).
The voltage error bars are from the standard error for digital meter readouts and the frequency was assumed to be known to around 1%. These allow for a spread (from the range-overtwo-method) in the best fit slope of ±4.85 × 10−16 V·s. From the regression, the uncertainty of
φ/e is ±.124V.
University of California, Los Angeles
Department of Physics & Astronomy
1
Physics 18-L
Modern Physics Lab
The Photoelectric Effect:
Measuring h/e
MacDonald, et al.
2.4.04
Figure 1: This plot depicts the linear behavior of the stopping voltage with varying frequency. The
red line is a linear regression of the data. The equation of the best fit and the physical significance is
detailed on the upper left.
Conclusion
The stopping voltage profile is indeed linear as expected in the range explored. Also, the electron
detection is almost instantaneous and the kinetic energy was frequency dependent, as one of
the lower frequency lines fails to produce any current. The intensity of the light was crudely
varied using a filter and only changed the current (not Vs ). This means that the kinetic energy
is independent of intensity.
The accepted value for h/e is 4.14 × 10−15 V·s. We determine the ratio to be (3.90 ±
0.49) × 10−15 V·s, resulting in an error of 5.7%. We are in good agreement with the known
value as it is well within our uncertainty. We further find that φ/e = (1.37 ± .12)V, or in
electron-volts, φ = (1.37 ± .12)eV. This is of the order of magnitude range for common metals.
Assuming the accepted value for h yields fcutoff = (3.31 ± .29) × 1014 Hz.
An underestimated h/e perhaps indicates an underestimated stopping voltage (at least at
higher frequencies). This seems plausible because these electrons are more energetic. Depending
on the geometry of the plates, some electrons may escape capture. A smaller effect may be due
the voltmeter “using” some electrons to measure the potential, thus resulting in a lower Vs .
Since only a portion of one decade was (sparsely) sampled, an improved experiment would
involve a more complete and wider range of frequencies.
University of California, Los Angeles
Department of Physics & Astronomy
2